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<?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:HSPGZ0AE.xml</dcterms:identifier>
<dcterms:creator>Harriot, Thomas</dcterms:creator>
<dcterms:title xml:lang="en">Mss. 6782</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/HSPGZ0AE</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="bundle" level="1" n="1">
<pb file="add_6782_f001" o="1" n="1"/>
<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">
AmazingLists of powers of two up to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mn>2</mn><mrow><mn>2</mn><mn>9</mn></mrow></msup></mrow></mstyle></math>, in standard denary (right) and in octonary (left).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head1" xml:space="preserve" xml:lang="lat">
Octonaria. Denaria.
<lb/>[<emph style="it">tr:
Octonary. Denary
</emph>]<lb/>
</head>
<pb file="add_6782_f001v" o="1v" n="2"/>
<pb file="add_6782_f002" o="2" n="3"/>
<pb file="add_6782_f002v" o="2v" n="4"/>
<pb file="add_6782_f003" o="3" n="5"/>
<pb file="add_6782_f003v" o="3v" n="6"/>
<pb file="add_6782_f004" o="4" n="7"/>
<pb file="add_6782_f004v" o="4v" n="8"/>
<pb file="add_6782_f005" o="5" n="9"/>
<pb file="add_6782_f005v" o="5v" n="10"/>
<pb file="add_6782_f006" o="6" n="11"/>
<pb file="add_6782_f006v" o="6v" n="12"/>
<pb file="add_6782_f007" o="7" n="13"/>
<pb file="add_6782_f007v" o="7v" n="14"/>
<pb file="add_6782_f008" o="8" n="15"/>
<pb file="add_6782_f008v" o="8v" n="16"/>
<pb file="add_6782_f009" o="9" n="17"/>
<pb file="add_6782_f009v" o="9v" n="18"/>
<pb file="add_6782_f010" o="10" n="19"/>
<pb file="add_6782_f010v" o="10v" n="20"/>
<pb file="add_6782_f011" o="11" n="21"/>
<pb file="add_6782_f011v" o="11v" n="22"/>
<pb file="add_6782_f012" o="12" n="23"/>
<pb file="add_6782_f012v" o="12v" n="24"/>
<pb file="add_6782_f013" o="13" n="25"/>
<pb file="add_6782_f013v" o="13v" n="26"/>
<pb file="add_6782_f014" o="14" n="27"/>
<pb file="add_6782_f014v" o="14v" n="28"/>
<pb file="add_6782_f015" o="15" n="29"/>
<pb file="add_6782_f015v" o="15v" n="30"/>
<pb file="add_6782_f016" o="16" n="31"/>
<pb file="add_6782_f016v" o="16v" n="32"/>
<pb file="add_6782_f017" o="17" n="33"/>
<pb file="add_6782_f017v" o="17v" n="34"/>
<pb file="add_6782_f018" o="18" n="35"/>
<pb file="add_6782_f018v" o="18v" n="36"/>
<pb file="add_6782_f019" o="19" n="37"/>
<pb file="add_6782_f019v" o="19v" n="38"/>
<pb file="add_6782_f020" o="20" n="39"/>
<pb file="add_6782_f020v" o="20v" n="40"/>
<pb file="add_6782_f021" o="21" n="41"/>
<pb file="add_6782_f021v" o="21v" n="42"/>
<pb file="add_6782_f022" o="22" n="43"/>
<pb file="add_6782_f022v" o="22v" n="44"/>
<pb file="add_6782_f023" o="23" n="45"/>
<pb file="add_6782_f023v" o="23v" n="46"/>
<pb file="add_6782_f024" o="24" n="47"/>
<pb file="add_6782_f024v" o="24v" n="48"/>
<pb file="add_6782_f025" o="25" n="49"/>
<pb file="add_6782_f025v" o="25v" n="50"/>
<pb file="add_6782_f026" o="26" n="51"/>
<pb file="add_6782_f026v" o="26v" n="52"/>
<pb file="add_6782_f027" o="27" n="53"/>
<div xml:id="echoid-div2" type="page_commentary" level="2" n="2">
<p>
<s xml:id="echoid-s3" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s3" xml:space="preserve">
A word square based on the words HENRICUS PRINCEPS FECIT (Prince Henry made it). <lb/>
The number 184,756 appears four times in the bottom right.
In each quarter of the square, there are 184,756 ways of reading HENRICUS PRINCEPS FECIT,
starting from the centre and ending at a corner. Thus there are 739,024 ways in total.
This number may have been written on the page but has now disappeared (see Add MS 6782, f. 28 for a similar calculation).
For the calculations leading to 184,756 see Add MS 6782, f. 57 and f. 58.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f027v" o="27v" n="54"/>
<pb file="add_6782_f028" o="28" n="55"/>
<div xml:id="echoid-div3" type="page_commentary" level="2" n="3">
<p>
<s xml:id="echoid-s5" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s5" xml:space="preserve">
A word square based on the words SILO PRINCEPS FECIT (Prince Henry made it). <lb/>
A <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>5</mn><mo>×</mo><mn>1</mn><mn>9</mn></mstyle></math> rectangular version of this arrangement, carved in stone, is to be found in the church of
San Juan Apostol y Evangelista in Santianes de Pravia, northern Spain,
commemorating Silo, king of Asturias (774 to 783). <lb/>
The number 12,780 appears four times in the bottom right hand corner of the page.
In each quarter of the square, there are 12,780 ways of reading SILO PRINCEPS FECIT,
starting from the centre and ending at a corner. Thus there are 51,480 ways in total.
For the calculations leading to 12,780 see Add MS 6782, f. 57 and f. 58.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f028v" o="28v" n="56"/>
<pb file="add_6782_f029" o="29" n="57"/>
<div xml:id="echoid-div4" type="page_commentary" level="2" n="4">
<p>
<s xml:id="echoid-s7" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s7" xml:space="preserve">
This folios shows calculations for solving the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>z</mi><mo>+</mo><mn>3</mn><mi>a</mi><mo>=</mo><mn>2</mn><mn>7</mn><mn>6</mn></mstyle></math>
(in modern notation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>3</mn><mi>x</mi><mo>=</mo><mn>2</mn><mn>7</mn><mn>6</mn></mstyle></math>).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f029v" o="29v" n="58"/>
<div xml:id="echoid-div5" type="page_commentary" level="2" n="5">
<p>
<s xml:id="echoid-s9" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s9" xml:space="preserve">
Triangular numbers arranged as dot patterns. <lb/>
The arrangement of Pascal's triangle on the right shows how each row is obtained
by summing two copies of the preceding row.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f030" o="30" n="59"/>
<div xml:id="echoid-div6" type="page_commentary" level="2" n="6">
<p>
<s xml:id="echoid-s11" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s11" xml:space="preserve">
The upper left quarter of the page contains two versions of Pascal's triangle, in different layouts. <lb/>
The upper right quarter shows the entries of the triangle
generated by successive multiplications of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo maxsize="1">)</mo></mstyle></math>. <lb/>
The lower right quarter shows the entries of the triangle
generated by successive multiplications of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo maxsize="1">)</mo></mstyle></math>. <lb/>
The lower left quarter demonstrates, following from the multiplications on the right,
that the entries in each row sum to a power of 2.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head2" xml:space="preserve" xml:lang="lat">
De combinationibus
<lb/>[<emph style="it">tr:
On combinations
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s13" xml:space="preserve">
To serve ones turne. <lb/>
</s>
<s xml:id="echoid-s14" xml:space="preserve">
To do one a sound turne.
</s>
</p>
<pb file="add_6782_f030v" o="30v" n="60"/>
<div xml:id="echoid-div7" type="page_commentary" level="2" n="7">
<p>
<s xml:id="echoid-s15" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s15" xml:space="preserve">
This page includes numerals from 1 to 9 written with a medieval form for '4',
and also with characters composed only of straight lines.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f031" o="31" n="61"/>
<div xml:id="echoid-div8" type="page_commentary" level="2" n="8">
<p>
<s xml:id="echoid-s17" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s17" xml:space="preserve">
The reference to 'Saxton's great map' is to Christopher Saxton's county maps for England and Wales,
published from 1579 onwards. <lb/>
Units of measurement:
a pase or pace (from Latin passus) was the length of a double stride, about 5 feet or 1.5 metres.
Thus one square pase was 25 square feet.
The Roman mile was 1000 pases. <lb/>
For some rough working for this page see Add MS 6788, f. 547v.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head3" xml:space="preserve">
An æstimable reckoning how many persons <lb/>
may inhabit the whole world.
</head>
<p>
<s xml:id="echoid-s19" xml:space="preserve">
Supositions.
</s>
<lb/>
<s xml:id="echoid-s20" xml:space="preserve">
[1]. The semiperimeter of a circle. 31,415,926
</s>
<lb/>
<s xml:id="echoid-s21" xml:space="preserve">
[2]. The semidiameter. 10,000,000
</s>
</p>
<p>
<s xml:id="echoid-s22" xml:space="preserve">
The compasse of the earth <lb/>
after the rate of 60 miles <lb/>
to a degree. 21,600 miles
</s>
</p>
<p>
<s xml:id="echoid-s23" xml:space="preserve">
Ergo: The halfe compasse. 10,300 miles
<sc>
The figure 10,300 is a copying error for 10,800.
The correct figure has been used in the subsequent calculations
</sc>
</s>
<lb/>
<s xml:id="echoid-s24" xml:space="preserve">
The semidiameter of the earth. 3,437 miles. 747 pases.
</s>
<lb/>
<s xml:id="echoid-s25" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mo>,</mo><mn>4</mn><mn>3</mn><mn>7</mn><mo>,</mo><mn>7</mn><mn>4</mn><mn>7</mn><mo>×</mo><mn>1</mn><mn>0</mn><mo>,</mo><mn>3</mn><mn>0</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>3</mn><mn>5</mn><mo>,</mo><mn>4</mn><mn>0</mn><mn>8</mn><mo>,</mo><mn>7</mn><mn>9</mn><mn>4</mn><mo>,</mo><mn>1</mn><mn>0</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>=</mo></mstyle></math> <lb/>
plano circuli. <lb/>
quod <reg norm="aequatur" type="abbr">æquat</reg>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>
<reg norm="superficies" type="abbr">superf</reg>: <lb/>
terræ et aquæ.
<lb/>[<emph style="it">tr:
a plane circle which equals <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> the surface of land and water.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s26" xml:space="preserve">
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mn>5</mn><mo>,</mo><mn>4</mn><mn>0</mn><mn>8</mn><mo>,</mo><mn>7</mn><mn>9</mn><mn>4</mn><mo>,</mo><mn>1</mn><mn>0</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>×</mo><mn>4</mn><mo>=</mo></mstyle></math> <lb/>
141,635,176,400,000
<reg norm="superficies" type="abbr">superf:</reg> maris et terræ.
<lb/>[<emph style="it">tr:
The surface of sea and land.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s27" xml:space="preserve">
70,817,588,200,000 =
<reg norm="superficies" type="abbr">superf:</reg> Terræ: vel maris.
<lb/>[<emph style="it">tr:
The surface of land or sea.
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s28" xml:space="preserve">
49,987 miles square in England
(<foreign xml:lang="lat">ut aliby</foreign> by Saxtons <lb/>
great map) after the rate of 60 miles to a degree <lb/>
including rivers &amp; all wastes.
<sc>
'aliby' is a copying error for 'alibi' (see Add MS 6788, f. 547v). <lb/>
</sc>
</s>
<lb/>
<s xml:id="echoid-s29" xml:space="preserve">
It lacketh but 13 miles of 50,000.
</s>
</p>
<p>
<s xml:id="echoid-s30" xml:space="preserve">
50,000 miles. 5,000,000 persons. supposed. <lb/>
1 mile. 100 persons. 70,817,588 miles. 7,081,758,800. persons in the earth.
</s>
</p>
<p>
 <s xml:id="echoid-s31" xml:space="preserve">[<emph style="it">Note:
This subcalculation gives 10,000 square pases per person,
converted next to 250,000 square feet, then to 5 and 8/11 acres.
</emph>]</s><lb/>
<s xml:id="echoid-s32" xml:space="preserve">
1,000,000 pp. 100 persons. <lb/>
10000 pp. 1 person. <lb/>
10000 [square pases] = <lb/>
250,000 [square feet] = <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mfrac><mrow><mn>8</mn></mrow><mrow><mn>1</mn><mn>1</mn></mrow></mfrac></mstyle></math> acres.
(note rivers &amp; waste included <lb/>
as above.)
</s>
</p>
<p>
<s xml:id="echoid-s33" xml:space="preserve">
6 men may stand in a pase square. <lb/>
</s>
<s xml:id="echoid-s34" xml:space="preserve">
Therefore 6,000,000 in one mile square. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mo>,</mo><mn>0</mn><mn>8</mn><mn>1</mn><mo>,</mo><mn>5</mn><mn>8</mn><mo>,</mo><mn>8</mn><mn>0</mn><mn>0</mn><mo>×</mo><mn>6</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>×</mo><mn>4</mn><mn>2</mn><mo>,</mo><mn>4</mn><mn>9</mn><mn>0</mn><mo>,</mo><mn>5</mn><mn>5</mn><mn>2</mn><mo>,</mo><mn>8</mn><mn>0</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>,</mo><mn>0</mn></mstyle></math>. <lb/>
</s>
<s xml:id="echoid-s35" xml:space="preserve">
The number of persons yt may stand on ye earth.
</s>
</p>
<pb file="add_6782_f031v" o="31v" n="62"/>
<div xml:id="echoid-div9" type="page_commentary" level="2" n="9">
<p>
<s xml:id="echoid-s36" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s36" xml:space="preserve">
For some of the calculations behind this page see Add MS 6788, f. 536, f. 537, f. 541.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head4" xml:space="preserve">
The issue from one man &amp; one woman in 240 yeares may be <lb/>
more then can inhabit the whole earth.
</head>
<p>
<s xml:id="echoid-s38" xml:space="preserve">
Supposing.
</s>
<lb/>
<s xml:id="echoid-s39" xml:space="preserve">
1. That the first man &amp; woman have a child every yeare <lb/>
one yeare male &amp; in other <emph style="super">yeare</emph> female
</s>
</p>
<p>
<s xml:id="echoid-s40" xml:space="preserve">
2. That the children when they are 20 yeares old &amp; upward <lb/>
do also every yeare beget a child one yeare male &amp; an <lb/>
other yeare female
</s>
</p>
<p>
<s xml:id="echoid-s41" xml:space="preserve">
3. That all are living at the end of 240 yeares. <lb/>
</s>
<s xml:id="echoid-s42" xml:space="preserve">
The number of <lb/>
males, 5,034,303,437. <lb/>
females, 5,034,303,437. <lb/>
persons, 10,068,606,874. <lb/>
(recconed in <foreign xml:lang="lat">charta</foreign> c)
[<emph style="it">Note:
Sheet c) is almost certainly Add MS 6788, f. 537,
where the same suppositions appear and this calculation is carried out,
but unfortunately the lettering of that page is obscured in the binding.
 </emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s43" xml:space="preserve">
That in 400 yeares upon the former suppositions <lb/>
there would be more men then can stand on the face <lb/>
of the whole <emph style="st">yeare</emph> earth.
</s>
</p>
<p>
<s xml:id="echoid-s44" xml:space="preserve">
(In <foreign xml:lang="lat">charta</foreign> db) I find that in 340 yeares they will make <lb/>
a number of 14 places.
[<emph style="it">Note:
Sheet db) is probably Add MS 6788, f. 541, but the lettering of that page is unfortunately obscured in the binding.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s45" xml:space="preserve">
Therefore in 400 yeares they will make a number of <lb/>
16 places which is more then can stand on the face of <lb/>
the earth (<foreign xml:lang="lat">ut versa pagina</foreign>.)
<sc>
'ut versa pagina' (as the other side of the page); see Add MS 6782, f. 31.
</sc>
</s>
</p>
<p>
<s xml:id="echoid-s46" xml:space="preserve">
How many persons have had being in 6000, yeares <lb/>
and in what roome they may stand.
</s>
</p>
<p>
<s xml:id="echoid-s47" xml:space="preserve">
Supposing <lb/>
1. That <emph style="st">in 40 yeares</emph> the world when it was replenished <lb/>
to the number of 7,000,000,000.
(<foreign xml:lang="lat">ut versa pagina</foreign>) <lb/>
they were alwayes one time with an other the same number.
<sc>
'ut versa pagina' (as the other side of the page); see Add MS 6782, f. 31.
</sc>
</s>
</p>
<p>
<s xml:id="echoid-s48" xml:space="preserve">
2. That in <emph style="st">every</emph> 40 yeares there is (one time with an other) as <lb/>
I have proved in an other page (&amp; agreeth with Nombers .26.64 <lb/>
a new generation.
</s>
</p>
<p>
<s xml:id="echoid-s49" xml:space="preserve">
Therefore in 6000 yeares there are 150 generations. <lb/>
</s>
<s xml:id="echoid-s50" xml:space="preserve">
Therefore there have been of persons in 6000 yeres <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>×</mo><mn>1</mn><mn>5</mn><mn>0</mn><mo>=</mo><mn>1</mn><mo>,</mo><mn>0</mn><mn>5</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn></mstyle></math>. persons. <lb/>
</s>
<s xml:id="echoid-s51" xml:space="preserve">
There being 50,000 miles square in England; <emph style="st">therefore</emph> there <lb/>
may stand in England 300,000,000,000. persons. <lb/>
</s>
<s xml:id="echoid-s52" xml:space="preserve">
The former number is three times greater, &amp; therefore <lb/>
there place of standing must be also 3 times greater then England.
</s>
</p>
<pb file="add_6782_f032" o="32" n="63"/>
<div xml:id="echoid-div10" type="page_commentary" level="2" n="10">
<p>
<s xml:id="echoid-s53" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s53" xml:space="preserve">
Another copy of the table shown on Add MS 6782, f. 63.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f032v" o="32v" n="64"/>
<pb file="add_6782_f033" o="33" n="65"/>
<div xml:id="echoid-div11" type="page_commentary" level="2" n="11">
<p>
<s xml:id="echoid-s55" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s55" xml:space="preserve">
This folio gives systematic lists of all combinations (without repetition) of the letters
<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>a</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>, <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>, <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><mi>f</mi></mstyle></math>.
In each case the combinations are listed as single letters, pairs, triples, and so on.
Combinations of the same size are listed alphabetically, thus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>d</mi></mstyle></math>, and so on.
(In the final column the triples are to be read downwards from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>e</mi><mi>f</mi></mstyle></math>
then back up from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>f</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi><mi>f</mi></mstyle></math>.) <lb/>
Numbers written to the right of the columns show the number of combinations in each part of the list.
Totals are given at the bottom. <lb/>
A table mid-left lists numbers of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mn>2</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mn>2</mn><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><mo>-</mo><mn>1</mn></mstyle></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>=</mo><mn>1</mn></mstyle></math>.
The latter are the numbers that appear as totals. <lb/>
A table lower left shows how combinations of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math> may be derived from those for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>
by adding <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> to the end of each of them and then also listing <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> as a singleton.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head5" xml:space="preserve">
Of combinations
</head>
<pb file="add_6782_f033v" o="33v" n="66"/>
<pb file="add_6782_f034" o="34" n="67"/>
<div xml:id="echoid-div12" type="page_commentary" level="2" n="12">
<p>
<s xml:id="echoid-s57" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s57" xml:space="preserve">
This folio gives systematic lists of all combinations (without repetition) of
<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>a</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>, <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>, <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><mi>f</mi></mstyle></math>.
In each case the combinations are listed as single letters, pairs, triples, nd so on.
Combinations of the same kind are listed alphabetically, thus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>d</mi></mstyle></math>, and so on.
Numbers written to the right of the columns show the number of combinations in each part of the list.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head6" xml:space="preserve">
Of combinations
</head>
<pb file="add_6782_f034v" o="34v" n="68"/>
<pb file="add_6782_f035" o="35" n="69"/>
<div xml:id="echoid-div13" type="page_commentary" level="2" n="13">
<p>
<s xml:id="echoid-s59" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s59" xml:space="preserve">
This folio shows combinations (without repetition) of <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>a</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>,
with each list constructed from the previous one.
Combinations of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>, for example, are found from combinations of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>
by adding <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> to each of them, together with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> as a singleton. <lb/>
Numbers to the right of each column show the number of combinations in each part of the list.
Totals are given at the bottom. <lb/>
Harriot uses the word 'complications' for combinations of more than one letter,
and 'simples' for single letters. <lb/>
At the end of the page Harriot reaches the conclusion that if null combinations are counted,
then the total number of combinations will be a power of 2.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head7" xml:space="preserve">
Of combinations
</head>
<p>
<s xml:id="echoid-s61" xml:space="preserve">
By this manner of construction &amp; <lb/>
generation of the variety of combinations <lb/>
or complications: <lb/>
these propositions are manifest:
</s>
</p>
<p>
<s xml:id="echoid-s62" xml:space="preserve">
The nomber of complications with the <lb/>
nomber of there simples: is double to the <lb/>
nomber of complications with there simples, <lb/>
of the next praecedent order: &amp; one more.
</s>
</p>
<p>
<s xml:id="echoid-s63" xml:space="preserve">
In any order of complications:
</s>
</p>
<p>
<s xml:id="echoid-s64" xml:space="preserve">
The nomber of <lb/>
Bynaryes Ternaryes Quaternaries &amp;c. <lb/>
is æquall to the nomber, in the precedent order, of: <lb/>
Binaryes &amp; Simples.
Ternary<emph style="super">es</emph>s &amp; Binaryes.
Quaternaryes &amp; Ternaryes. &amp; c.
</s>
</p>
<p>
<s xml:id="echoid-s65" xml:space="preserve">
Hereby is <emph style="st">[???]</emph> also manifest, the reason &amp; order <lb/>
of the construction of the table of combinations by nombers <lb/>
which <emph style="st">followeth</emph> <emph style="super">is set downe</emph>
in an other paper.
</s>
</p>
<p>
<s xml:id="echoid-s66" xml:space="preserve">
In some cases rationall or negative <lb/>
is <emph style="st">[???]</emph> added: as of the species <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. question may be made whether an other <lb/>
thing hath acte upon. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. or both <lb/>
that is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. or neither.
</s>
</p>
<p>
<s xml:id="echoid-s67" xml:space="preserve">
And such negatives may be understode <lb/>
of all the rest.
</s>
<s xml:id="echoid-s68" xml:space="preserve">
And thus the sommes of <lb/>
every order wilbe one more; and there <lb/>
progression wilbe. 2. 4. 8. 16. 32. &amp;c.
</s>
</p>
<p>
<s xml:id="echoid-s69" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, or not. <lb/>
</s>
<s xml:id="echoid-s70" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. or not. <lb/>
</s>
<s xml:id="echoid-s71" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>. or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>. or not. <lb/>
&amp;c.
</s>
</p>
<pb file="add_6782_f035v" o="35v" n="70"/>
<pb file="add_6782_f036" o="36" n="71"/>
<div xml:id="echoid-div14" type="page_commentary" level="2" n="14">
<p>
<s xml:id="echoid-s72" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s72" xml:space="preserve">
Columns 1 to 10 of Pascal's triangle, continued in the lower half of the page as far as column 18. <lb/>
The table is continued to column 24 on Add MS 6782, f. 37.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head8" xml:space="preserve">
Of combinations
</head>
<p>
<s xml:id="echoid-s74" xml:space="preserve">
These nombers are also the <lb/>
nombers which are used for the <lb/>
extraction of rootes.
</s>
</p>
<pb file="add_6782_f036v" o="36v" n="72"/>
<pb file="add_6782_f037" o="37" n="73"/>
<div xml:id="echoid-div15" type="page_commentary" level="2" n="15">
<p>
<s xml:id="echoid-s75" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s75" xml:space="preserve">
Pascal's triangle as far as column 24, continued from Add MS 6782, f. 36.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head9" xml:space="preserve">
Of combinations
</head>
<pb file="add_6782_f037v" o="37v" n="74"/>
<pb file="add_6782_f038" o="38" n="75"/>
<div xml:id="echoid-div16" type="page_commentary" level="2" n="16">
<p>
<s xml:id="echoid-s77" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s77" xml:space="preserve">
On this page, as on Add MS 6782, f. 36, Harriot recognizes the usefulness of the triangular numbers
both for calculating numbers of combinations and for extraction of roots. <lb/>
In the note in the bottom right hand corner of the page, Harriot mentions Boethius, Jordanus, and Maurolico
as writers on figurate numbers.
His source for both Boethius and Jordanus was probably Jacques Lefevre d'Etaples (Jacob Faber Satuplensis),
<emph style="it">Epitome, compendiosaque introductio in libros arithmeticos diui Severini Boetij</emph> (1503, 1522),
which includes a comparison of the 'De instituione' of Boethius with the 'De arithmetica' of Jordanus.
His source for Maurolico was the <emph style="it">Arithmeticorum libri duo</emph> (1575),
which he cited several times elsewhere.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head10" xml:space="preserve">
Of combinations
</head>
<p>
<s xml:id="echoid-s79" xml:space="preserve">
A Generall rule to get the mayne summe of all the complications <lb/>
of any Nomber of Species without the table of combinations.
</s>
</p>
<p>
<s xml:id="echoid-s80" xml:space="preserve">
According to the nomber of species: understand as many termes <lb/>
to be gotten in continuall proportion or progression, beginning at <lb/>
a unite &amp; making every terme double to his precedent: the
<emph style="super">double of the</emph> last <lb/>
terme lesse by a unite is the summe desired. or the somme of the <lb/>
progression.
</s>
</p>
<p>
<s xml:id="echoid-s81" xml:space="preserve">
As for example.
</s>
<s xml:id="echoid-s82" xml:space="preserve">
I wold know all the complications <lb/>
of 6. species. together with the nomber of the simples. <lb/>
</s>
<s xml:id="echoid-s83" xml:space="preserve">
the sixth terme of such a progression I spake of, <lb/>
is 32.
</s>
<s xml:id="echoid-s84" xml:space="preserve">
<emph style="st">therefore</emph>
<emph style="super">The double lesse by a unite is</emph> 63,
<emph style="st">is</emph> the summe of all the <emph style="st">the</emph> <lb/>
complications with the nomber of simples which were <lb/>
sought.
</s>
<lb/>
<s xml:id="echoid-s85" xml:space="preserve">
If the number of species be greate; the last terme <lb/>
desired is to be gotten by <emph style="super">the</emph> rule of progression in <lb/>
arithmeticke.
</s>
<lb/>
<s xml:id="echoid-s86" xml:space="preserve">
The reason of the rule is easily to be conceaved <lb/>
out of the particular constructions in an other <lb/>
paper annexed.
<sc>
The 'other paper' referred to here is probably Add MS 6782, f. 331.
</sc>
</s>
</p>
<p>
<s xml:id="echoid-s87" xml:space="preserve">
A Generall methode for the particular summes of <lb/>
complications :
</s>
</p>
<p>
<s xml:id="echoid-s88" xml:space="preserve">
As for example of 6.
</s>
<s xml:id="echoid-s89" xml:space="preserve">
<emph style="st">I would know all</emph> first in 6 there are <lb/>
6 diverse simple species.
</s>
<s xml:id="echoid-s90" xml:space="preserve">
Then I wold know how many <lb/>
complications of 2 wilbe found in 6; also how many of 3. &amp; 4. &amp;c.
</s>
</p>
<p>
<s xml:id="echoid-s91" xml:space="preserve">
The Theoreme for the rule is this: <lb/>
</s>
</p>
<p>
<s xml:id="echoid-s92" xml:space="preserve">
As 2 hath in proportion to the second nomber from the nomber <lb/>
of species towardes a unite: so hath the nomber of variety of unites <lb/>
to the somme of the complications of 2.
</s>
<lb/>
<s xml:id="echoid-s93" xml:space="preserve">
And as 3 hath in proportion to the third nomber from the nomber <lb/>
of species towardes a unite: so hath the nomber of <emph style="st">variety of</emph> compli-<lb/>
cations of 2, last gotten; to the somme of the complications of 3. <lb/>
</s>
<s xml:id="echoid-s94" xml:space="preserve">
&amp; so forth, <emph style="st">generally</emph> as wilbe manifest by the example following.
</s>
</p>
<p>
<s xml:id="echoid-s95" xml:space="preserve">
An example for 20.
</s>
</p>
<p>
<s xml:id="echoid-s96" xml:space="preserve">
The practice is playne. <lb/>
</s>
<s xml:id="echoid-s97" xml:space="preserve">
The theoreme is to be <lb/>
demonstrated out of <lb/>
Boetius or Maurolicus, <lb/>
&amp; I thinke Jordanus. <lb/>
</s>
<s xml:id="echoid-s98" xml:space="preserve">
by the doctrine of genera- <lb/>
ting triangular nombers <lb/>
&amp; of triangular, piramidal. <lb/>
&amp; of piramidal, triangle- <lb/>
pyramidal &amp;c. <lb/>
</s>
<s xml:id="echoid-s99" xml:space="preserve">
And is worth the noting <lb/>
for some other respects <lb/>
especially <emph style="st">of generating</emph> <lb/>
for getting the nomber <lb/>
of [¿]complicity[?] that <lb/>
belongs to any dignityes <lb/>
for extracting there roote; <lb/>
seeing those nombers are <lb/>
the very same.
</s>
</p>
<pb file="add_6782_f038v" o="38v" n="76"/>
<pb file="add_6782_f039" o="39" n="77"/>
<div xml:id="echoid-div17" type="page_commentary" level="2" n="17">
<p>
<s xml:id="echoid-s100" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s100" xml:space="preserve">
By 'transpositions' Harriot means what we would now call permutations.
His 'single variations' are what we would now call cyclic permutations.
For simple diagrams illustrating cyclic permutations see Add MS 6782, f. 43v and f. 225v. <lb/>
On this folio he lists all possible permutations of <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>a</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>,
and begins a list for <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>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head11" xml:space="preserve">
Of Transpositions
</head>
<p>
<s xml:id="echoid-s102" xml:space="preserve">
Single variations.
</s>
</p>
<p>
<s xml:id="echoid-s103" xml:space="preserve">
They are <lb/>
so many as <lb/>
there are <lb/>
species.
</s>
</p>
<p>
<s xml:id="echoid-s104" xml:space="preserve">
The nomber of transpositions of any nomber of species being given: The nomber <lb/>
of transpositions of the next nomber of species, is a nomber that riseth of there <lb/>
multiplication.
</s>
</p>
<p>
<s xml:id="echoid-s105" xml:space="preserve">
For: suppose the nomber of transpositions of 3 species, that is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, to be 6. The <lb/>
next nomber to be transposed is 4. which let be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>.
</s>
<s xml:id="echoid-s106" xml:space="preserve">
Now <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, in respect of <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math> hath foure places. that is he may be next after <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>: after <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>: after <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>: or <lb/>
before <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>.
</s>
<s xml:id="echoid-s107" xml:space="preserve">
So many places it hath with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mi>b</mi></mstyle></math>, &amp; the rest of the 6.
</s>
<s xml:id="echoid-s108" xml:space="preserve">
Therefore 4 times <lb/>
6, which is 24. is the nomber of transpositions of 4 species.
</s>
<s xml:id="echoid-s109" xml:space="preserve">
The like reason <lb/>
is of all others that <emph style="st">may</emph> follow
<foreign xml:lang="lat">in infinitum</foreign>.
</s>
</p>
<p>
<s xml:id="echoid-s110" xml:space="preserve">
Transpositions of:
</s>
</p>
<p>
<s xml:id="echoid-s111" xml:space="preserve">
Number of <lb/>
Termes of <lb/>
variations
</s>
</p>
<p>
<s xml:id="echoid-s112" xml:space="preserve">
Sume of the <lb/>
species totall
</s>
</p>
<pb file="add_6782_f039v" o="39v" n="78"/>
<pb file="add_6782_f040" o="40" n="79"/>
<div xml:id="echoid-div18" type="page_commentary" level="2" n="18">
<p>
<s xml:id="echoid-s113" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s113" xml:space="preserve">
This small table shows all the combinations and permutations of up to 7 letters.
The figures in the column under 7, for example, show all the combinations of 7 single letters (7),
all the combinations of 2 letters (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mo>×</mo><mn>6</mn><mo>=</mo><mn>4</mn><mn>2</mn></mstyle></math>),
of 3 letters (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mo>×</mo><mn>6</mn><mo>×</mo><mn>5</mn><mo>=</mo><mn>2</mn><mn>1</mn><mn>0</mn></mstyle></math>), and so on. <lb/>
Lines drawn between columns show how figures in a given column are obtained
from those in the preceding column.
The figures in column 7, for example, are obtained from those in column 6 by multiplying by 7.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head12" xml:space="preserve">
combinations &amp; transpositions together.
</head>
<pb file="add_6782_f040v" o="40v" n="80"/>
<div xml:id="echoid-div19" type="page_commentary" level="2" n="19">
<p>
<s xml:id="echoid-s115" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s115" xml:space="preserve">
Tables showing the most likely sums on 5 dice, or on 6 dice (totals only). <lb/>
The left hand table is a frequency table for the sums on five dice,
constructed by adding copies of the columns of table (v) from Add MS 6782, f. 40
(just as table (v) there was constructed from table (iv). <lb/>
The two right hand columns are a frequency table for the sums on six dice.
Here the individual columns that make up the sum have not been written down but
the additions have been carried out in the working at the bottom of the page.
In the table itself Harriot has written only the totals. <lb/>
As on Add MS 6782, f. 40, the most likely sums are marked with crosses.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f041" o="41" n="81"/>
<div xml:id="echoid-div20" type="page_commentary" level="2" n="20">
<p>
<s xml:id="echoid-s117" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s117" xml:space="preserve">
The two words 'on dice' in the title are written in Harriot's Algonquin alphabet
(see Add MS 6782, f. 337). <lb/>
The two tables at the top of the page show the sums that can be obtained on
(i) two dice (sums ranging from 2 to 12) or (ii) three dice (sums ranging from 3 to 18). <lb/>
The three tables in the lower part of the page are frequency tables for the sums on
(iii) two dice; (iv) three dice ; (v) four dice. <lb/>
Tables (iii) and (iv) can be calculated directly from (i) and (ii).
However, the layout shows that (iv) can also be calculated by taking copies of the totals from (iii),
staggering their starting position, and then adding;
this is equivalent to taking the totals from (iii)
and then adding 1, 2, 3, 4, 5, 6, in turn to represent the throw of the third dice. <lb/>
For each of (iii), (iv), and (v) the most likely sums are marked with a small cross.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head13" xml:space="preserve">
Of combinations &amp; transpositions of the numbers [on diz]
</head>
<pb file="add_6782_f041v" o="41v" n="82"/>
<pb file="add_6782_f042" o="42" n="83"/>
<div xml:id="echoid-div21" type="page_commentary" level="2" n="21">
<p>
<s xml:id="echoid-s119" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s119" xml:space="preserve">
Another version of the tables from the lower half of Add MS 6782, f. 81.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f042v" o="42v" n="84"/>
<div xml:id="echoid-div22" type="page_commentary" level="2" n="22">
<p>
<s xml:id="echoid-s121" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s121" xml:space="preserve">
The word 'diz' (dice) is written at the top of the page in Harriot's Algonquin alphabet
(see Add MS 6782, f. 337.)
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s123" xml:space="preserve">
[diz]
</s>
</p>
<pb file="add_6782_f043" o="43" n="85"/>
<pb file="add_6782_f043v" o="43v" n="86"/>
<pb file="add_6782_f044" o="44" n="87"/>
<div xml:id="echoid-div23" type="page_commentary" level="2" n="23">
<p>
<s xml:id="echoid-s124" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s124" xml:space="preserve">
This folio quotes some text from Girolam Cardano, <emph style="it">Opus novum de proportionibus</emph>
(1570), page 187, Proposition 170. The copy is in an unknown hand.
The table below the text is exactly as given by Cardano.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head14" xml:lang="lat">
Cardanus de proportionibus. prop. 170.
</head>
<p xml:lang="lat">
<s xml:id="echoid-s126" xml:space="preserve">
Ut autem habeas numeros singulorum ordinum, in quavis multitudino, deducito <lb/>
numerum ordinis a primo, &amp; divide per numerum ordinis ipsius reliquum, <lb/>
et illud quod proventi, ducito in numerum maximum praecedentis ordinis, <lb/>
et habebis numerum quaesitum. </s>
<s xml:id="echoid-s127" xml:space="preserve">
Velut si sint undecim, volo scire breviter numeros, <lb/>
qui fiunt ex variatione trium. </s>
<s xml:id="echoid-s128" xml:space="preserve">
Primum deduco pro secundo ordine 1 ex 11 fit 10, <lb/>
divido per 2 numerum ordinis, exit 5, duco in 11 fit 55 numerus secundi ordinis. Inde <lb/>
detraho 2, qui est numerus differentiae ordinis totij a primo ex 11, relinquitur 9, <lb/>
divido 9 per 3 numerum ordinis exit 3, duco 3 in 55 numerum secundi fit 165, <lb/>
numerus totij ordinis. </s>
<s xml:id="echoid-s129" xml:space="preserve">
Similiter volo numerum variatione quatuor, <lb/>
dedco 3 differentiam 4 a primo ordine ab 11 relinquitur 8, divido 8 per 4 <lb/>
numero ordinis, exit 2, duco 2 in 195 fit 330, numeri quarti ordinis. </s><lb/>
<s xml:id="echoid-s130" xml:space="preserve">
Similiter pro quinto detraho 4 differentiam a primo ordine, relinquitur 7, <lb/>
divido per 5 numerum ordinis exit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mfrac><mrow><mn>2</mn></mrow><mrow><mn>5</mn></mrow></mfrac></mstyle></math>, duco in 330 numero praecedentis <lb/>
ordine, fit 462 numerus quinti ordinis.
<lb/>[<emph style="it">tr:
As moreover you have the numbers of a single row, of any size,
you can derive the numbers of each place from the first,
and divide by the number of the place, and what arises multiply by the greatest number
of the preceding place,and you will have the number sought.
Thus if there are eleven [objects],
I want to know quickly the number that arises for three choices. First, for the second place,
I take 1 from 11 which makes 10, I divide by 2, the number of the place, there comes out 5,
I multiply by 11 to make 55, the number of the second place. Next I subtract 2,
which is the number of the difference of all the places from the first, from 11, there remains 9,
I divide 9 by 3,
the number of the place, there comes out 3, I multiply 3 by 55, the second number, to make 165,
the number in the third place. Similarly if I want the number for four choices, I take 3,
the difference of 4 from the first place, from 11, there is left 8, I divide 8 by 4,
the number of the place, there comes out 2, I multiply 2 by 165 to make 330,
the number of the fourth place.
Similarly for the fifth I subtract 4, the difference from the first place, there remains 7, I divide by 5,
the number of the place, there comes out <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mfrac><mrow><mn>2</mn></mrow><mrow><mn>5</mn></mrow></mfrac></mstyle></math>, I multiply by 330,
the number of the previous place, to make 462, the number of the fifth place.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f044v" o="44v" n="88"/>
<pb file="add_6782_f045" o="45" n="89"/>
<div xml:id="echoid-div24" type="page_commentary" level="2" n="24">
<p>
<s xml:id="echoid-s131" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s131" xml:space="preserve">
This folio appears to deal with 6 dice numbered as follows:
</s>
<lb/>
<s xml:id="echoid-s132" xml:space="preserve">
first die: 0 0 0 0 0 1 <lb/>
second die: 0 0 0 0 0 2 <lb/>
third die: 0 0 0 0 0 3 <lb/>
fourth die: 0 0 0 0 0 4 <lb/>
fifth die: 0 0 0 0 0 5 <lb/>
sixth die: 0 0 0 0 0 6
</s>
<lb/>
<s xml:id="echoid-s133" xml:space="preserve">
The tables show the possible outcomes of throwing the first, then the first and the second,
then the first and the second and the third, and so on.
In the table for six dice, for example, we see that combinations with one 0 and five other numbers
can appear in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mo>×</mo><mn>6</mn><mo>=</mo><mn>3</mn><mn>0</mn></mstyle></math> ways (since any of the five zeros can appear in any of the six positions). <lb/>
Squeezed below and between the main tables are frequency tables showing how many times each sum can appear.
For six dice, for example, we see that the total 3 can arise in two ways (as 1 + 2 or as 0 + 3)
giving 3,750 possibilities in total.
In each of these tables the number of possibilites is summed, giving the appropriate power of 6 in each case.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f045v" o="45v" n="90"/>
<pb file="add_6782_f046" o="46" n="91"/>
<pb file="add_6782_f046v" o="46v" n="92"/>
<pb file="add_6782_f047" o="47" n="93"/>
<pb file="add_6782_f047v" o="47v" n="94"/>
<pb file="add_6782_f048" o="48" n="95"/>
<pb file="add_6782_f048v" o="48v" n="96"/>
<div xml:id="echoid-div25" type="page_commentary" level="2" n="25">
<p>
<s xml:id="echoid-s135" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s135" xml:space="preserve">
Combinations of quantities generated by multiplication. <lb/>
The letters <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> stand for
<foreign xml:lang="lat">pondus</foreign> (weight),
<foreign xml:lang="lat">magnitudo</foreign> (magnitude),
<foreign xml:lang="lat">figura</foreign> (area),
<foreign xml:lang="lat">situs</foreign> (place),
<foreign xml:lang="lat">altitudo</foreign> (altitude)
(see Add MS 6786, f. 291).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f049" o="49" n="97"/>
<div xml:id="echoid-div26" type="page_commentary" level="2" n="26">
<p>
<s xml:id="echoid-s137" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s137" xml:space="preserve">
The tables on this folio appear to have been begun at the top left
but then re-started and continued along the right-hand edge. <lb/>
The tables are calculated in turn for 1, 2, 3, 4, 5, 6 throws of a die. <lb/>
Take, for example, the fourth table, for four throws of a die. <lb/>
The first row indicates that the combination 1111 can occur in only one way. <lb/>
The next two rows indicate how many ways only 1 and 2 can occur, distributed as either
3 + 1 (thus, 1112, 1121, 1211, 2111, 2221, 2212, 2122, 1222), that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mo>×</mo><mn>4</mn><mo>=</mo><mn>8</mn></mstyle></math> ways in total,
or as
2 + 2 (thus, 1122, 1212, 1221, 2112, 2121, 2211), that is, 3 + 3 = 6 ways in total.
These two calculations are shown in full on Add MS 6782, f. 50v. <lb/>
The fourth row indicates how many ways only 1, 2, and 3 can occur,
with any one of them appearing twice (thus 1123, 1132, 1212, 3112, ...), that is,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>3</mn><mo>=</mo><mn>3</mn><mn>6</mn></mstyle></math> ways in total.
Further details of the calculation are shown on Add MS 6782, f. 50v. <lb/>
The fifth and final row indicates how many ways 1, 2, 3, 4 can appear, that is,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>4</mn><mo>=</mo><mn>2</mn><mn>4</mn></mstyle></math> ways in total. <lb/>
All the other tables are calculated in a similar way.
Several of the calculations can be seen on Add MS 6782, f. 50v. <lb/>
Below the line (still reading the page sideways) are two further tables;
for the continuation of these, see Add MS 6782, f. 50. <lb/>
Harriot also includes some brief notes to explain how the tables have been derived.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s139" xml:space="preserve">
11112 variatur per:
<lb/>[<emph style="it">tr:
11112 may be varied by:
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s140" xml:space="preserve">
conversionem, ut
22221.
<lb/>[<emph style="it">tr:
conversion, as 22221.
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s141" xml:space="preserve">
transpositionem, <lb/>
11112, <lb/>
11121, <lb/>
11211, <lb/>
12111, <lb/>
21111
<lb/>[<emph style="it">tr:
transposition, <lb/>
11112, <lb/>
11121, <lb/>
11211, <lb/>
12111, <lb/>
21111
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s142" xml:space="preserve">
coniugationum ut <emph style="st">supra</emph> <lb/>
sunt 2 ex 6; sunt 15<emph style="super">ies</emph>
<lb/>[<emph style="it">tr:
conjugation, as there are 2 out of 6, there are 15 ways
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f049v" o="49v" n="98"/>
<pb file="add_6782_f050" o="50" n="99"/>
<div xml:id="echoid-div27" type="page_commentary" level="2" n="27">
<p>
<s xml:id="echoid-s143" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s143" xml:space="preserve">
On this folio Harriot uses the totals he has arrived at on Add MS 6783, f. 49,
but now extends the calculations to all six numbers on the die. <lb/>
As for Add MS 6783, f. 49, we will once again examine the fourth table. <lb/>
The first row indicates that a given number can be appear four times in just one way;
but the given number can be chosen in six ways, so there are <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>×</mo><mn>6</mn><mo>=</mo><mn>6</mn></mstyle></math> such combinations in total. <lb/>
The second row indicates that any two numbers can occur together, with 3 of one and 1 of the other, in 8 ways
(as calculated on Add MS 6783, f. 49,);
but two numbers can be chosen from six in 15 ways (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>6</mn><mo>×</mo><mn>5</mn></mrow><mrow><mn>2</mn><mo>×</mo><mn>1</mn></mrow></mfrac></mstyle></math>),
so there are <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>8</mn><mo>×</mo><mn>5</mn><mo>=</mo><mn>1</mn><mn>2</mn><mn>0</mn></mstyle></math> such combinations in total. <lb/>
The third row indicates that any two numbers can occur together, with 2 of one and 2 of the other, in 6 ways
(as calculated on Add MS 6783, f. 49,);
but two numbers can be chosen from six in 15 ways (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>6</mn><mo>×</mo><mn>5</mn></mrow><mrow><mn>2</mn><mo>×</mo><mn>1</mn></mrow></mfrac></mstyle></math>),
so there are <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mo>×</mo><mn>1</mn><mn>5</mn><mo>=</mo><mn>9</mn><mn>0</mn></mstyle></math> such combinations in total. <lb/>
The fourth row indicates that any three numbers can occur together in 36 ways
(as calculated on Add MS 6783, f. 49,);
but three numbers can be chosen from six in 20 ways (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>6</mn><mo>×</mo><mn>5</mn><mo>×</mo><mn>4</mn></mrow><mrow><mn>3</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>1</mn></mrow></mfrac></mstyle></math>),
so there are <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mn>6</mn><mo>×</mo><mn>2</mn><mn>0</mn><mo>=</mo><mn>7</mn><mn>2</mn><mn>0</mn></mstyle></math> such combinations in total. <lb/>
The fifth row indicates that any four numbers can occur together in 24 ways
(as calculated on Add MS 6783, f. 49,);
but four numbers can be chosen from six in 15 ways (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>6</mn><mo>×</mo><mn>5</mn><mo>×</mo><mn>4</mn><mo>×</mo><mn>3</mn></mrow><mrow><mn>4</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>1</mn></mrow></mfrac></mstyle></math>),
so there are <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>4</mn><mo>×</mo><mn>1</mn><mn>5</mn><mo>=</mo><mn>3</mn><mn>6</mn><mn>0</mn></mstyle></math> such combinations in total. <lb/>
As a check on the calculations, Harriot has calculated powers of 6 on the same page.
It is easily seen from this that each table includes all the possibilities for that number of throws.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f050v" o="50v" n="100"/>
<div xml:id="echoid-div28" type="page_commentary" level="2" n="28">
<p>
<s xml:id="echoid-s145" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s145" xml:space="preserve">
Calculations for Add MS 6782, f. 97.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f051" o="51" n="101"/>
<pb file="add_6782_f051v" o="51v" n="102"/>
<pb file="add_6782_f052" o="52" n="103"/>
<pb file="add_6782_f052v" o="52v" n="104"/>
<pb file="add_6782_f053" o="53" n="105"/>
<pb file="add_6782_f053v" o="53v" n="106"/>
<pb file="add_6782_f054" o="54" n="107"/>
<pb file="add_6782_f054v" o="54v" n="108"/>
<pb file="add_6782_f055" o="55" n="109"/>
<div xml:id="echoid-div29" type="page_commentary" level="2" n="29">
<p>
<s xml:id="echoid-s147" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s147" xml:space="preserve">
For the definition of binomes of the third kind, see Add MS 6782, f. 267. <lb/>
On this page, Harriot shows that the cube of a binome of the third kind is again a binome of the third kind.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head15" xml:space="preserve" xml:lang="lat">
De cubo binomij 3<emph style="super">i</emph>
<lb/>[<emph style="it">tr:
On the cube of a third binome
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s149" xml:space="preserve">
<reg norm="binomij" type="abbr">bin</reg>. 3.
<lb/>[<emph style="it">tr:
a binome of the third kind.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s150" xml:space="preserve">
<reg norm="binomij" type="abbr">bin</reg>. 1.
<lb/>[<emph style="it">tr:
a binome of the first kind.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s151" xml:space="preserve">
Ergo cubus: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>5</mn><mn>4</mn><mn>0</mn><mn>8</mn></mrow></msqrt><mo>+</mo><msqrt><mrow><mn>5</mn><mn>4</mn><mn>0</mn><mn>0</mn></mrow></msqrt></mstyle></math>. <reg norm="binomij" type="abbr">bin</reg>. 3.
<reg norm="differentia" type="abbr">diff</reg>.
<reg norm="quadrati" type="abbr">quad</reg>: <lb/>
8. cubus.
<lb/>[<emph style="it">tr:
Therefore the cube <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>5</mn><mn>4</mn><mn>0</mn><mn>8</mn></mrow></msqrt><mo>+</mo><msqrt><mrow><mn>5</mn><mn>4</mn><mn>0</mn><mn>0</mn></mrow></msqrt></mstyle></math> is a binome of the third kind;
the difference of the squares <lb/>
is 8, a cube.</emph>]<lb/>
</s>
<lb/>
</p>
<pb file="add_6782_f055v" o="55v" n="110"/>
<pb file="add_6782_f056" o="56" n="111"/>
<div xml:id="echoid-div30" type="page_commentary" level="2" n="30">
<p>
<s xml:id="echoid-s152" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s152" xml:space="preserve">
For the definition of binomes of the fourth kind, see Add MS 6782, f. 267. <lb/>
On this page, Harriot shows that the cube of a binome of the fourth kind is again a binome of the fourth kind.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head16" xml:space="preserve" xml:lang="lat">
De cubo binomij 4<emph style="super">i</emph>
<lb/>[<emph style="it">tr:
On the cube of a fourth binome
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s154" xml:space="preserve">
<reg norm="binomij" type="abbr">bin</reg>. 4.
<lb/>[<emph style="it">tr:
a binome of the fourth kind.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s155" xml:space="preserve">
<reg norm="binomij" type="abbr">bin</reg>. 1.
<lb/>[<emph style="it">tr:
a binome of the first kind.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s156" xml:space="preserve">
Ergo cubus <reg norm="binomij" type="abbr">bin</reg>. 4.
<lb/>[<emph style="it">tr:
Therefore the cube is a binome of the fourth kind;
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s157" xml:space="preserve">
512. <reg norm="differentia" type="abbr">dra</reg> <reg norm="quadrati" type="abbr">quad</reg>: <lb/>
cub
<lb/>[<emph style="it">tr:
512. the difference of the squares, is a cube.</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s158" xml:space="preserve">
Aliter
<lb/>[<emph style="it">tr:
Another way
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s159" xml:space="preserve">
Ergo cubus <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mn>2</mn><mn>0</mn><mo>+</mo><msqrt><mrow><mn>5</mn><mn>1</mn><mn>7</mn><mo>,</mo><mn>8</mn><mn>8</mn><mn>8</mn></mrow></msqrt></mstyle></math> <lb/>
Ut supra.
<lb/>[<emph style="it">tr:
Therefore the cube is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mn>2</mn><mn>0</mn><mo>+</mo><msqrt><mrow><mn>5</mn><mn>1</mn><mn>7</mn><mo>,</mo><mn>8</mn><mn>8</mn><mn>8</mn></mrow></msqrt></mstyle></math>. As above.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f056v" o="56v" n="112"/>
<pb file="add_6782_f057" o="57" n="113"/>
<div xml:id="echoid-div31" type="page_commentary" level="2" n="31">
<p>
<s xml:id="echoid-s160" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s160" xml:space="preserve">
The table at the top left shows the number of pathways from centre to corner
for squares of size <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>×</mo><mn>1</mn></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>1</mn><mo>×</mo><mn>2</mn><mn>1</mn></mstyle></math>. For a <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mo>×</mo><mn>5</mn></mstyle></math> square, for instance,
there are 20 pathways to each corner, and so 80 in all.
(There is an error in the calculation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>9</mn><mo>×</mo><mn>9</mn></mstyle></math> squares, where there are 70 pathways to each corner,
making a total of 280 in all, not 240.) <lb/>
Below the table are the numbers of pathways for the squares on Add MS 6782, f. 28 and f. 27,
SILO PRINCEPS FECIT (17 letters, 51,480 pathways) and HENRICUS PRINCEPS FECIT (21 letters, 739,024 pathways). <lb/>
The calculations down the right hand side of the page show the multiplication
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>1</mn><mo>×</mo><mn>1</mn><mn>2</mn><mo>×</mo><mn>1</mn><mn>3</mn><mo>×</mo><mo>…</mo><mo>×</mo><mn>2</mn><mn>1</mn></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s162" xml:space="preserve">
Silo princeps fecit (17) 51,480
<lb/>[<emph style="it">tr:
Prince Silo made it
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s163" xml:space="preserve">
Jacobus
</s>
<lb/>
<s xml:id="echoid-s164" xml:space="preserve">
Henricus princeps fecit (21) 739,024
<lb/>[<emph style="it">tr:
Prince Henry made it
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s165" xml:space="preserve">
Carolus princeps fecit (20)
<lb/>[<emph style="it">tr:
Prince Charles made it
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f057v" o="57v" n="114"/>
<div xml:id="echoid-div32" type="page_commentary" level="2" n="32">
<p>
<s xml:id="echoid-s166" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s166" xml:space="preserve">
The upper half of the page shows calculations of the number of pathways through a quarter square
for up to 9 letters. <lb/>
The calculation in the lower half of the page arrives at the total 705,432.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f058" o="58" n="115"/>
<div xml:id="echoid-div33" type="page_commentary" level="2" n="33">
<p>
<s xml:id="echoid-s168" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s168" xml:space="preserve">
This folios shows a version of Pascal's triangle, with the numbers 1, 2, 6, 20, 70, ...
along the diagonal emphasized. <lb/>
Here the numbers represent the number of pathways through a square with up to 144 cells,
from the starting point (marked 0) to any cell in the grid.
The numbers along the diagonal show the number of pathways from corner to corner,
as required on Add MS 6782, f. 27 and f. 28.
The numbers 12,870 and 184,756 from f. 28 and f. 27 appear on this diagonal,
as does the number 705,432 calculated on Add MS 6782, f. 57v.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s170" xml:space="preserve">
<reg norm="examinatur" type="abbr">examinat</reg>.
<lb/>[<emph style="it">tr:
examined
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f058v" o="58v" n="116"/>
<pb file="add_6782_f059" o="59" n="117"/>
<pb file="add_6782_f059v" o="59v" n="118"/>
<pb file="add_6782_f060" o="60" n="119"/>
<div xml:id="echoid-div34" type="page_commentary" level="2" n="34">
<p>
<s xml:id="echoid-s171" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s171" xml:space="preserve">
Quarter squares, completed with numbers.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f060v" o="60v" n="120"/>
<pb file="add_6782_f061" o="61" n="121"/>
<pb file="add_6782_f061v" o="61v" n="122"/>
<pb file="add_6782_f062" o="62" n="123"/>
<pb file="add_6782_f062v" o="62v" n="124"/>
<pb file="add_6782_f063" o="63" n="125"/>
<div xml:id="echoid-div35" type="page_commentary" level="2" n="35">
<p>
<s xml:id="echoid-s173" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s173" xml:space="preserve">
The table on this folio shows the same information as Add MS 6782, f. 58,
but now in the form of one quarter of a word square.
The 'letters' in each cell are the upper entries, in slightly heavier ink.
The lower entry in each cell is the number of pathways to that square, starting from the top left hand corner. <lb/>
Grid squares along the diagonal have been slightly shaded for emphasis.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f063v" o="63v" n="126"/>
<pb file="add_6782_f064" o="64" n="127"/>
<div xml:id="echoid-div36" type="page_commentary" level="2" n="36">
<p>
<s xml:id="echoid-s175" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s175" xml:space="preserve">
Squares of size <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>×</mo><mn>1</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mo>×</mo><mn>3</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mo>×</mo><mn>5</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mo>×</mo><mn>7</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>9</mn><mo>×</mo></mstyle></math>,
completed with numbers.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f064v" o="64v" n="128"/>
<pb file="add_6782_f065" o="65" n="129"/>
<pb file="add_6782_f065v" o="65v" n="130"/>
<pb file="add_6782_f066" o="66" n="131"/>
<pb file="add_6782_f066v" o="66v" n="132"/>
<pb file="add_6782_f067" o="67" n="133"/>
<div xml:id="echoid-div37" type="page_commentary" level="2" n="37">
<p>
<s xml:id="echoid-s177" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s177" xml:space="preserve">
In this page. Harriot begins by constructing a table of the interest paid after seven years
on a capital sum of £<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math> at an annual rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>. <lb/>
The first row shows the total if the interest is paid yearly (7 payments). <lb/>
The second row shows the total if interest is paid twice a year (14 payments). <lb/>
The third row shows the total if interest is paid three times a year (21 payments). <lb/>
From here, Harriot immediately generalizes to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> payments per year.
He then (implicitly) allows <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> to become very large, indeed infinitely large,
so that the fractions <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><mi>n</mi><mi>n</mi></mrow></mfrac></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mo maxsize="1">(</mo><mi>n</mi><mo>-</mo><mn>2</mn><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo maxsize="1">)</mo><mi>n</mi></mrow><mrow><mi>n</mi><mi>n</mi><mi>n</mi></mrow></mfrac></mstyle></math>, ... can all be taken to be 1.
Harriot then substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mn>1</mn><mn>0</mn></mstyle></math> to obtain the interest on £100 at a rate of <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>,
paid continuously over seven years. The total comes to £201 7 shillings and 6 pence,
plus a further fraction that he estimates is not quite <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>7</mn></mrow><mrow><mn>1</mn><mn>0</mn><mn>0</mn></mrow></mfrac></mstyle></math> pence.
(There were 20 shillings (s) to £1, and 12 pence (d) to 1 shillling.) <lb/>
This page combines the calculations on the nearby f. 68
(interest on £<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> at an annual rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>, paid at decreasing intervals)
with those f. 69
(interest on £100 at an annual rate of <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> paid at decreasing intervals and taken to a limit.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head17" xml:space="preserve">
Interest upon interest for 7 yeares.
</head>
<p>
<s xml:id="echoid-s179" xml:space="preserve">
The sum of interest upon interest <lb/>
continually for every instant <emph style="super">in</emph> seven <lb/>
yeares with the principall of 100£ <lb/>
after the rate of 10 in the 100 for <lb/>
a yeare.
</s>
</p>
<p>
<s xml:id="echoid-s180" xml:space="preserve">
201£ + 7s + 6d + <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>0</mn><mn>6</mn><mn>2</mn><mn>0</mn><mn>5</mn><mi>a</mi></mrow><mrow><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn></mrow></mfrac></mstyle></math> <lb/>
not <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>7</mn></mrow><mrow><mn>1</mn><mn>0</mn><mn>0</mn></mrow></mfrac></mstyle></math>
</s>
</p>
<pb file="add_6782_f067v" o="67v" n="134"/>
<pb file="add_6782_f068" o="68" n="135"/>
<div xml:id="echoid-div38" type="page_commentary" level="2" n="38">
<p>
<s xml:id="echoid-s181" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s181" xml:space="preserve">
Below the rough work crossed out at the top are calculations of interest on £<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math>
at an annual rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>. <lb/>
The first table shows the interest paid every year for seven years.
The words <foreign xml:lang="lat">continue proportionales</foreign> (continued proportionals)
indicate that each row is obtained by multiplication from the previous row.
The multiplier from each row to the next is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>b</mi></mrow></mfrac><mo maxsize="1">)</mo></mstyle></math>. <lb/>
The second table shows a similar calculation but now interest is added twice yearly
and the multiplier is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>b</mi></mrow></mfrac><mo maxsize="1">)</mo></mstyle></math>.
The table shows only the first four entries and then the total after 7 years. <lb/>
The third table repeats the calculation but now interest is added three times yearly
and the multiplier is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn><mi>b</mi></mrow></mfrac><mo maxsize="1">)</mo></mstyle></math>.
The table shows only the first three entries and then the total after 7 years.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s183" xml:space="preserve">
yeres
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s184" xml:space="preserve">
continue proportionales
<lb/>[<emph style="it">tr:
continued proportionals
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f068v" o="68v" n="136"/>
<pb file="add_6782_f069" o="69" n="137"/>
<div xml:id="echoid-div39" type="page_commentary" level="2" n="39">
<p>
<s xml:id="echoid-s185" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s185" xml:space="preserve">
After the rough work crossed out at the top,
the table show the calculation of interest on £100 at an annual rate of <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>,
paid every year for five years. <lb/>
The next section is crossed out but the reader is referred by asterisk to an expression lower down the page.
Here Harriot has written a general formula for the interest paid on £100
at an annual rate of <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> after <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> years. <lb/>
In the next calculation Harriot has assumed that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> is infinitely large,
so that the fractions <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><mi>n</mi><mi>n</mi></mrow></mfrac></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mo maxsize="1">(</mo><mi>n</mi><mo>-</mo><mn>2</mn><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo maxsize="1">)</mo><mi>n</mi></mrow><mrow><mi>n</mi><mi>n</mi><mi>n</mi></mrow></mfrac></mstyle></math>, ... can all be taken to be 1.
Thus he obtains the interest on £100 at an annual rate of <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> paid continuously. <lb/>
In calculating the sum, Harriot has drawn a vertical line that cuts off the calculation after 6 terms.
He notes underneath that £<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn><mn>0</mn></mrow></mfrac><mo>=</mo><mn>4</mn></mstyle></math>d (4 pence), that £<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mn>4</mn><mn>0</mn><mn>0</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mstyle></math>d,
and that the sum of all the remaining terms will not make up as much as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>2</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mstyle></math>d.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s187" xml:space="preserve">
yeares
</s>
</p>
<p>
<s xml:id="echoid-s188" xml:space="preserve">
The summe of interest upon interest <lb/>
continually for every instant <lb/>
the whole yeare with the principall <lb/>
of 100£ after the rate of 10 in ye 100 <lb/>
for the yeare.
</s>
</p>
<p>
<s xml:id="echoid-s189" xml:space="preserve">
not <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>2</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mstyle></math> d
</s>
</p>
<pb file="add_6782_f069v" o="69v" n="138"/>
<pb file="add_6782_f070" o="70" n="139"/>
<div xml:id="echoid-div40" type="page_commentary" level="2" n="40">
<p>
<s xml:id="echoid-s190" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s190" xml:space="preserve">
The tables in the first row show the calculation of interest on £<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>
at an annual rate of <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> for four years. <lb/>
The tables in the second row show the calculation of interest on £<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>
at an annual rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mn>0</mn></mrow></mfrac></mstyle></math> for four years. <lb/>
The tables in the third row show the calculation of interest on £<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>
at an annual rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn><mn>0</mn></mrow></mfrac></mstyle></math> for four years. <lb/>
The tables in the last row show the calculation of interest on £<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>
at an annual rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mi>c</mi></mrow></mfrac></mstyle></math> for seven years. <lb/>
The words <foreign xml:lang="lat">continue proportionales</foreign> (continued proportionals)
next to the final table indicate that each row is obtained by multiplication from the previous row.
The multiplier from each row to the next is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>c</mi></mrow></mfrac><mo maxsize="1">)</mo></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f070v" o="70v" n="140"/>
<pb file="add_6782_f071" o="71" n="141"/>
<pb file="add_6782_f071v" o="71v" n="142"/>
<pb file="add_6782_f072" o="72" n="143"/>
<pb file="add_6782_f072v" o="72v" n="144"/>
<div xml:id="echoid-div41" type="page_commentary" level="2" n="41">
<p>
<s xml:id="echoid-s192" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s192" xml:space="preserve">
A list of texts and page numbers.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s194" xml:space="preserve">
Compendium Avicennæ. 17. <lb/>
Brevi Breviarum pro Baconis <lb/>
ad <emph style="super">[???]</emph> [???]. 95. <lb/>
Verbum abreviatum [???] <lb/>
Raymondi de Leone Vividi. 26A. <lb/>
Secretum secretorum naturæ de lazuli <lb/>
lapidis philosphonum. 285 <lb/>
Tractatus trium Bretonum Br. Baconis. <lb/>
Epistola prima. 293. <lb/>
Secunda 301. <lb/>
Tertia 314. <lb/>
Specatum secretorum. 387.
</s>
</p>
<pb file="add_6782_f073" o="73" n="145"/>
<div xml:id="echoid-div42" type="page_commentary" level="2" n="42">
<p>
<s xml:id="echoid-s195" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s195" xml:space="preserve">
The choice of numbers on this page suggests that it might have been written in the yeasr 1599?
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s197" xml:space="preserve">
The denary Arithmetick which is in common use <lb/>
doth expresse nombers of figures in a continuall progression <lb/>
of which a unit is the first; the second is ten &amp; may <lb/>
be termed as in a algebra a roote; the third is a hundred &amp; <lb/>
may be termed a square. &amp;c.
</s>
</p>
<pb file="add_6782_f073v" o="73v" n="146"/>
<pb file="add_6782_f074" o="74" n="147"/>
<div xml:id="echoid-div43" type="page_commentary" level="2" n="43">
<p>
<s xml:id="echoid-s198" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s198" xml:space="preserve">
At the beginning of this set of sheets Harriot has written: 'Waste papers of figurate nombers'.
They are waste only in the sense that they contain rough working. At the same time,
they show Harriot attempting something highly original, namely,
finding formulae for sequences of figurate numbers.
In modern terms, we would say he is fitting third-, fourth- or fifth-degree polynomials
to numerical sequences. <lb/>
At the top is the sequence 1, 5, 14, 30, 55, ... of sums of squares
(or of square-pyramidal numbers, see Add MS 6782, f. 155).
Just below that, Harriot has written the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>9</mn><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>1</mn><mi>a</mi><mo>+</mo><mn>2</mn><mi>a</mi></mstyle></math>, which, it seems,
is his first attempt to find a formula for the numbers in the sequence multiplied by 24
(that is, 24, 120, 336, ...). Putting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>1</mn></mstyle></math> gives <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mo>+</mo><mn>9</mn><mo>+</mo><mn>1</mn><mn>1</mn><mo>+</mo><mn>2</mn></mstyle></math>, as required.
Putting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn></mstyle></math>, however, gives <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mo>×</mo><mn>1</mn><mn>6</mn><mo>+</mo><mn>9</mn><mo>×</mo><mn>8</mn><mo>+</mo><mn>1</mn><mn>1</mn><mo>+</mo><mn>4</mn><mo>+</mo><mn>2</mn><mo>×</mo><mn>4</mn><mo>=</mo><mn>1</mn><mn>5</mn><mn>2</mn></mstyle></math>, which is too large.
This calculation can be seen displayed vertically just below the formula.
Harriot notes that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mo>×</mo><mn>2</mn><mn>4</mn><mo>=</mo><mn>1</mn><mn>4</mn><mn>4</mn></mstyle></math> falls short of this total by 8. <lb/>
Similar trial and error calculations appear on this and several pages that follow.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head18" xml:space="preserve">
Waste papers. <lb/>
of figurate <lb/>
nombers.
</head>
<pb file="add_6782_f074v" o="74v" n="148"/>
<div xml:id="echoid-div44" type="page_commentary" level="2" n="44">
<p>
<s xml:id="echoid-s200" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s200" xml:space="preserve">
Trials similar to those on f. 74, but now for the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mi>a</mi><mo>+</mo><mn>1</mn><mn>1</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>6</mn><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>,
represented by the coefficients 6, 11, 6, 1. This is evaluated for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn></mstyle></math>,
giving <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mn>6</mn><mn>0</mn><mo>=</mo><mn>2</mn><mn>4</mn><mo>×</mo><mn>1</mn><mn>5</mn></mstyle></math>;
for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>5</mn></mstyle></math>, giving <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>6</mn><mn>8</mn><mn>0</mn><mo>=</mo><mn>2</mn><mn>4</mn><mo>×</mo><mn>7</mn><mn>0</mn></mstyle></math>; and (lower right) for for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn></mstyle></math>,
giving <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>2</mn><mn>0</mn><mo>=</mo><mn>2</mn><mn>4</mn><mo>×</mo><mn>5</mn></mstyle></math>.
It seems Harriot is still trying to fit the sequence 1, 5, 14, 30, ...,
which appears again alongside the coefficients 6, 11, 6, 1 on f. 76v.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f075" o="75" n="149"/>
<pb file="add_6782_f075v" o="75v" n="150"/>
<pb file="add_6782_f076" o="76" n="151"/>
<pb file="add_6782_f076v" o="76v" n="152"/>
<div xml:id="echoid-div45" type="page_commentary" level="2" n="45">
<p>
<s xml:id="echoid-s202" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s202" xml:space="preserve">
Further calculations with the coefficients 6, 11, 6, 1 (see f. 74v), now for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>4</mn></mstyle></math>,
giving <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>8</mn><mn>4</mn><mn>0</mn><mo>=</mo><mn>2</mn><mn>4</mn><mo>×</mo><mn>3</mn><mn>5</mn></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f077" o="77" n="153"/>
<div xml:id="echoid-div46" type="page_commentary" level="2" n="46">
<p>
<s xml:id="echoid-s204" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s204" xml:space="preserve">
On this folio Harriot appears to be searchng for a formula for the sequence 1, 6, 20, 50, ...
of sums of square-pyramidal numbers (see Add MS 6782, f. 155), or rather,
for those numbers multiplied 24, that is, 24, 144, 480, ... . <lb/>
Examples on the left hand side of the page test the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>8</mn><mi>a</mi><mo>+</mo><mn>4</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>0</mn><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>2</mn><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>
(note that 8 + 4 + 10 + 2 = 24). This is evaluated for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn></mstyle></math>, giving 144 (as required)
and for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn></mstyle></math>, giving 492 (too large). <lb/>
Examples in the bottom left hand corner rearrange the same coefficients in different orders:
(2, 10, 8, 4), (8, 10, 2, 4), (8, 2, 10, 4), etc. <lb/>
In examples further to the right, the fifth-degree polynomial <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>4</mn><mi>a</mi><mo>+</mo><mn>5</mn><mn>0</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>3</mn><mn>5</mn><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>0</mn><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>,
represented by the coefficients 24, 50, 35, 10, 1, is evaluated for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn></mstyle></math>, giving 640,
and for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn></mstyle></math> giving 2520.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f077v" o="77v" n="154"/>
<div xml:id="echoid-div47" type="page_commentary" level="2" n="47">
<p>
<s xml:id="echoid-s206" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s206" xml:space="preserve">
This page shows further attempts, as on f. 77, to find coefficients that deliver
24 (when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>1</mn></mstyle></math>), 144 (when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn></mstyle></math>), and 480 (when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn></mstyle></math>).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f078" o="78" n="155"/>
<pb file="add_6782_f078v" o="78v" n="156"/>
<pb file="add_6782_f079" o="79" n="157"/>
<pb file="add_6782_f079v" o="79v" n="158"/>
<pb file="add_6782_f080" o="80" n="159"/>
<div xml:id="echoid-div48" type="page_commentary" level="2" n="48">
<p>
<s xml:id="echoid-s208" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s208" xml:space="preserve">
On this folio Harriot is seraching for a formula for the sequence 1, 6, 15, 28, ... of hexagonal numbers
(see Add MS 6782, f. 157), or rather, for those numbers multiplied by 24, that is, 24, 144, 360, .... <lb/>
Note that some of the coefficients on this page are negative.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f080v" o="80v" n="160"/>
<pb file="add_6782_f081" o="81" n="161"/>
<div xml:id="echoid-div49" type="page_commentary" level="2" n="49">
<p>
<s xml:id="echoid-s210" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s210" xml:space="preserve">
On this folio Harriot is searching for a formula for the sequence 1, 5, 12, 22, ... of pentagonal numbers
(see f. Add MS 6782, 156), or rather, for those numbers multiplied by 24, that is, 24, 120, 288, ....
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f081v" o="81v" n="162"/>
<div xml:id="echoid-div50" type="page_commentary" level="2" n="50">
<p>
<s xml:id="echoid-s212" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s212" xml:space="preserve">
This folio shows many examples of fifth-degree polynomials, as represented by their coefficients,
evaluated for <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>a</mi><mo>=</mo><mn>3</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>5</mn></mstyle></math>. At top left, for example,
the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mn>0</mn><mi>a</mi><mo>+</mo><mn>2</mn><mn>3</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>5</mn><mn>2</mn><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>3</mn><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> is evaluated for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn></mstyle></math>, giving 840.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f082" o="82" n="163"/>
<div xml:id="echoid-div51" type="page_commentary" level="2" n="51">
<p>
<s xml:id="echoid-s214" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s214" xml:space="preserve">
Like f. 81v, this folio shows examples of fifth-degree polynomials, as represented by their coefficients,
evaluated for <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>a</mi><mo>=</mo><mn>3</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>5</mn></mstyle></math>. At top left, for example,
the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>8</mn><mi>a</mi><mo>+</mo><mn>4</mn><mn>5</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>4</mn><mn>0</mn><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>5</mn><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>2</mn><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> is evaluated for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn></mstyle></math>, giving 3240.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f082v" o="82v" n="164"/>
<pb file="add_6782_f083" o="83" n="165"/>
<div xml:id="echoid-div52" type="page_commentary" level="2" n="52">
<p>
<s xml:id="echoid-s216" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s216" xml:space="preserve">
On this folio Harriot appears to be searching a formula for the sequence 1, 7, 27, 77, ...
of sums of sums of square-pyramidal numbers (see Add MS 6782, f. 155), or rather,
for those numbers multiplied by 120, that is, 120, 840, 3240, ....
For ths he needs polynmials of the fifth degree.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f083v" o="83v" n="166"/>
<pb file="add_6782_f084" o="84" n="167"/>
<div xml:id="echoid-div53" type="page_commentary" level="2" n="53">
<p>
<s xml:id="echoid-s218" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s218" xml:space="preserve">
The first reference in the heading is to Michael Stifel, <emph style="it">Arithmetica integra</emph> (1544),
page 15.
For Stifel, a diagonal number was obtained by multiplying the first two entries of a Pythagorean triple.
The diagonal number corresponding to the triple (3, 4, 5), for example, is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mo>×</mo><mn>4</mn><mo>=</mo><mn>1</mn><mn>2</mn></mstyle></math>.
Stifel also defined Pythagorean triples by the ratio of the two shorter sides, in this case <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math>.
He was able to write out two lists, or orders, of triples,
one with the shorter side odd (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn><mn>2</mn></mrow><mrow><mn>5</mn></mrow></mfrac></mstyle></math>, and so on),
the other with the shorter side even (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn><mn>5</mn></mrow><mrow><mn>8</mn></mrow></mfrac></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>3</mn><mn>5</mn></mrow><mrow><mn>1</mn><mn>2</mn></mrow></mfrac></mstyle></math>, and so on).
Stifel claimed that all possible triples were included in these two orders. <lb/>
The second reference in the heading, possibly added a little later, is to Johannes Praetorius (Johann Richter),
<emph style="it">Problema, quod iubet ex quatuor rectis lineis datis quadrilaterum fieri,
quod sit in circulo</emph> (1598). On the final page,
Praetorius discusses the problem of constructing cyclic quadrilaterals with rational sides. <lb/>
<lb/>
Harriot sets out to disprove Stifel's claim, by demonstrating the existence of new orders of triples. <lb/>
His first order (ordo. 1.) is the same as Stifel's first order.
The triples are set out in three columns with differences calculated between rows.
This allows Harriot to extrapolate forwards, but also backwards to a starting triple (1, 0, 1). <lb/>
The second order (ordo. 2.) is the same as Stifel's second order.
Again the triples are set out in three columns with differences calculated between rows.
As for the first order this allows Harriot to extrapolate backwards to a starting triple (4, 3, 5).
This is the first triple of the first order with the first two entries interchanged.
Perhaps this gave Harriot the idea of interchanging other pairs.
Thus he begins a third and new order (ordo. 3. novus) with (12, 5, 8),
which is the second triple from the first order with the first two entries interchanged.
This order immediately contains (20, 21, 29), which was not included in either of Stifel's orders.
The fourth order begins with (15, 8, 17),
which is the first triple from the second order with the first two entries interchanged.
And so on.
<lb/>
By the end of the page, Harriot has six orders, with differences in the left column of 2, 4, 8, 6, 10, 12,
respectively.
This seems to suggest to him a more systematic method of displaying the orders,
which he goes on to do on the next page.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head19" xml:space="preserve" xml:lang="lat">
Examinatur Stifelius <lb/>
de numeris diagonalibus. pa. 15 <lb/>
et prætorius. pag. ult
<lb/>[<emph style="it">tr:
An examination of Stifel on diagonal numbers, page 15, and Praetorius, last page.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s220" xml:space="preserve">
ordo. 1. <lb/>
pythag.
<lb/>[<emph style="it">tr:
Order 1, Pythagorean
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s221" xml:space="preserve">
ord. 2. <lb/>
<reg norm="Platonic" type="abbr">platon</reg>.
<lb/>[<emph style="it">tr:
Order 2, Platonic.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s222" xml:space="preserve">
hoc est.
<lb/>[<emph style="it">tr:
that is:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s223" xml:space="preserve">
Dixit quod rationes omni <lb/>
laterum sunt in istis <lb/>
duobus ordinibus.
<lb/>[<emph style="it">tr:
He said that all the ratios of sides are in these two orders.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s224" xml:space="preserve">
Ego dico quod non.
<lb/>[<emph style="it">tr:
I say that is not so.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s225" xml:space="preserve">
ordines sunt alij <lb/>
infiniti.
</s>
<s xml:id="echoid-s226" xml:space="preserve">
Ut per <lb/>
sequentia patet.
<lb/>[<emph style="it">tr:
There are infinitely many other orders. As is clear from what follows.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s227" xml:space="preserve">
ordo. 1. <lb/>
ordo. 2. <lb/>
ordo. 3. novus.
<lb/>[<emph style="it">tr:
Order 1. <lb/>
Order 2. <lb/>
Order 3, new.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s228" xml:space="preserve">
Melior est dispositio ordinum <lb/>
in alijs chartis sequentibus.
<lb/>[<emph style="it">tr:
The arrangement of orders is better in the other sheets following.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f084v" o="84v" n="168"/>
<div xml:id="echoid-div54" type="page_commentary" level="2" n="54">
<p>
<s xml:id="echoid-s229" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s229" xml:space="preserve">
This folio gives a list of the hypotenuses that have been discovered on the previous page,
with the differences between them.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head20" xml:space="preserve" xml:lang="lat">
Hypotenusorum progressio
<lb/>[<emph style="it">tr:
A progression of the hypotenuses
</emph>]<lb/>
</head>
<pb file="add_6782_f085" o="85" n="169"/>
<div xml:id="echoid-div55" type="page_commentary" level="2" n="55">
<p>
<s xml:id="echoid-s231" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s231" xml:space="preserve">
On this folio the orders discovered on the previous sheet (f. 84) are listed systematically.
Each new order begins with the second triple from the previous order,
with the first two entries interchanged. <lb/>
At the bottom of the page, Harriot notes the starting differences for each order. <lb/>
He has also written the enigmatic note
</s>
<lb/>
<quote xml:lang="lat">
Hic sunt omnes primi sed hic omnes non sunt primi
</quote>
<lb/>
<s xml:id="echoid-s232" xml:space="preserve">
which was greatly to confuse his friend Nathaniel Torporley when he came across it some years later. <lb/>
For discussion of this and the surrounding sheets see Tanner 1977.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s234" xml:space="preserve">
1.)
</s>
</p>
<p>
<s xml:id="echoid-s235" xml:space="preserve">
1) ordo.
<lb/>[<emph style="it">tr:
order 1
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s236" xml:space="preserve">
Et sic in cæteribus in infinitum.
<lb/>[<emph style="it">tr:
And so on for the rest indefinitely.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s237" xml:space="preserve">
Hic sunt omnes primi <lb/>
sed hic omnes non sunt primi.
<lb/>[<emph style="it">tr:
Here are all the primes but here not all are prime.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s238" xml:space="preserve">
Nota <lb/>
prima Differentia ordinis <lb/>
primi. 2. 4. sub dupla. <lb/>
Secundi. 4. 12. tripla <lb/>
Tertij. 6. 12. Dupla <lb/>
Quarti. 8. 24. tripla <lb/>
Quinti. 10. 20. Dupla <lb/>
Sexti. 12. 36. tripla <lb/>
Septimi. 14. 28. Dupla <lb/>
Octavi. 16. 48. tripla <lb/>
&amp;c in infinitum.
<lb/>[<emph style="it">tr:
Note<lb/>
First differences of the order <lb/>
First double <lb/>
Second triple <lb/>
Third double <lb/>
Fourth triple <lb/>
Fifth double <lb/>
Sixth triple <lb/>
Seventh double <lb/>
Eighth triple <lb/>
etc. indefinitely
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f085v" o="85v" n="170"/>
<pb file="add_6782_f086" o="86" n="171"/>
<div xml:id="echoid-div56" type="page_commentary" level="2" n="56">
<p>
<s xml:id="echoid-s239" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s239" xml:space="preserve">
Harriot continues his orders 1 to 4 of Pythagorean triples from f. 85.
In each case the orders are continued as far as the hypotenuse closest to 1105.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s241" xml:space="preserve">
2.) continuationes
<lb/>[<emph style="it">tr:
continuations
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s242" xml:space="preserve">
1) ordinis
<lb/>[<emph style="it">tr:
order 1
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s243" xml:space="preserve">
recte
<lb/>[<emph style="it">tr:
correct
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f086v" o="86v" n="172"/>
<pb file="add_6782_f087" o="87" n="173"/>
<div xml:id="echoid-div57" type="page_commentary" level="2" n="57">
<p>
<s xml:id="echoid-s244" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s244" xml:space="preserve">
Harriot continues his orders 5 to 7 of Pythagorean triples from f. 85, and adds order 8.
In each case the orders are continued until the hypotenuse is equal to or greater 1105.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s246" xml:space="preserve">
3.)
</s>
</p>
<p>
<s xml:id="echoid-s247" xml:space="preserve">
5 ordo.
<lb/>[<emph style="it">tr:
order 5
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s248" xml:space="preserve">
recte
<lb/>[<emph style="it">tr:
correct
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f087v" o="87v" n="174"/>
<pb file="add_6782_f088" o="88" n="175"/>
<div xml:id="echoid-div58" type="page_commentary" level="2" n="58">
<p>
<s xml:id="echoid-s249" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s249" xml:space="preserve">
Harriot lists his orders 9 to 12 of Pythagorean triples,
continuing in each case until the hypotenuse is greater than 1105.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s251" xml:space="preserve">
4)
</s>
</p>
<pb file="add_6782_f088v" o="88v" n="176"/>
<pb file="add_6782_f089" o="89" n="177"/>
<div xml:id="echoid-div59" type="page_commentary" level="2" n="59">
<p>
<s xml:id="echoid-s252" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s252" xml:space="preserve">
Harriot lists his orders 13 to 22 of Pythagorean triples,
continuing in each case until the hypotenuse is equal to or greater than 1105.
Unfortunately an error in the last step of order 19 has led him to miss one of the triples ending in 1105:
the final set of differences should have been 38, 84, 84,
leading to the triple (817, 744, 1105).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s254" xml:space="preserve">
<emph style="super">5</emph>.)
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s255" xml:space="preserve">
&amp;c. in infinitum.
<lb/>[<emph style="it">tr:
etc. indefinitely.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f089v" o="89v" n="178"/>
<pb file="add_6782_f090" o="90" n="179"/>
<div xml:id="echoid-div60" type="page_commentary" level="2" n="60">
<p>
<s xml:id="echoid-s256" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s256" xml:space="preserve">
On this folio, Harriot demonstrates both geometrically and arithmetically that
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mrow><mn>8</mn><mn>5</mn></mrow><mn>2</mn></msup></mrow><mo>=</mo><mrow><msup><mrow><mn>8</mn><mn>4</mn></mrow><mn>2</mn></msup></mrow><mo>+</mo><mrow><msup><mrow><mn>1</mn><mn>2</mn></mrow><mn>2</mn></msup></mrow><mo>+</mo><mrow><msup><mn>4</mn><mn>2</mn></msup></mrow><mo>+</mo><mrow><msup><mn>3</mn><mn>2</mn></msup></mrow></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s258" xml:space="preserve">
one square æquall to many. <lb/>
whence <lb/>
To devide one square into <lb/>
many
</s>
</p>
<pb file="add_6782_f090v" o="90v" n="180"/>
<pb file="add_6782_f091" o="91" n="181"/>
<div xml:id="echoid-div61" type="page_commentary" level="2" n="61">
<p>
<s xml:id="echoid-s259" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s259" xml:space="preserve">
An investigation into Pythagorean triples with hypotenuse 1105.
Harriot obtains several such triples by multiplication of triples already known.
Those marked 'supra +' or '+ supra' duplicate others earlier in the list. <lb/>
Three triples are identified by Harriot as prime, that is, with no common factors,
that is, (47, 1104, 1105), (264, 1073, 1105), (576, 943, 1105).
An error in his 19th order, on Add MS 6782, f. 89, has led him to miss a fourth, (817, 744, 1105).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s261" xml:space="preserve">
supra +
<lb/>[<emph style="it">tr:
as above
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s262" xml:space="preserve">
primi
<lb/>[<emph style="it">tr:
prime
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f091v" o="91v" n="182"/>
<pb file="add_6782_f092" o="92" n="183"/>
<pb file="add_6782_f092v" o="92v" n="184"/>
<pb file="add_6782_f093" o="93" n="185"/>
<pb file="add_6782_f093v" o="93v" n="186"/>
<pb file="add_6782_f094" o="94" n="187"/>
<pb file="add_6782_f094v" o="94v" n="188"/>
<pb file="add_6782_f095" o="95" n="189"/>
<pb file="add_6782_f095v" o="95v" n="190"/>
<div xml:id="echoid-div62" type="page_commentary" level="2" n="62">
<p>
<s xml:id="echoid-s263" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s263" xml:space="preserve">
Successive halving of 90 degrees.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f096" o="96" n="191"/>
<div xml:id="echoid-div63" type="page_commentary" level="2" n="63">
<p>
<s xml:id="echoid-s265" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s265" xml:space="preserve">
Successive halving of 90 degrees, continued from the previous page (f. 95v).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f096v" o="96v" n="192"/>
<pb file="add_6782_f097" o="97" n="193"/>
<pb file="add_6782_f097v" o="97v" n="194"/>
<pb file="add_6782_f098" o="98" n="195"/>
<pb file="add_6782_f098v" o="98v" n="196"/>
<pb file="add_6782_f099" o="99" n="197"/>
<pb file="add_6782_f099v" o="99v" n="198"/>
<pb file="add_6782_f100" o="100" n="199"/>
<pb file="add_6782_f100v" o="100v" n="200"/>
<pb file="add_6782_f101" o="101" n="201"/>
<pb file="add_6782_f101v" o="101v" n="202"/>
<div xml:id="echoid-div64" type="page_commentary" level="2" n="64">
<p>
<s xml:id="echoid-s267" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s267" xml:space="preserve">
A double-page calculation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>2</mn></mrow></msqrt></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f102" o="102" n="203"/>
<pb file="add_6782_f102v" o="102v" n="204"/>
<pb file="add_6782_f103" o="103" n="205"/>
<pb file="add_6782_f103v" o="103v" n="206"/>
<pb file="add_6782_f104" o="104" n="207"/>
<pb file="add_6782_f104v" o="104v" n="208"/>
<div xml:id="echoid-div65" type="page_commentary" level="2" n="65">
<p>
<s xml:id="echoid-s269" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s269" xml:space="preserve">
A calculation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>2</mn></mrow></msqrt></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f105" o="105" n="209"/>
<pb file="add_6782_f105v" o="105v" n="210"/>
<pb file="add_6782_f106" o="106" n="211"/>
<div xml:id="echoid-div66" type="page_commentary" level="2" n="66">
<p>
<s xml:id="echoid-s271" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s271" xml:space="preserve">
64 digits of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>2</mn></mrow></msqrt></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f106v" o="106v" n="212"/>
<div xml:id="echoid-div67" type="page_commentary" level="2" n="67">
<p>
<s xml:id="echoid-s273" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s273" xml:space="preserve">
A check on the first few digits of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>2</mn></mrow></msqrt></mstyle></math> by multiplication.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f107" o="107" n="213"/>
<div xml:id="echoid-div68" type="page_commentary" level="2" n="68">
<p>
<s xml:id="echoid-s275" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s275" xml:space="preserve">
A half sheet, very worn, and darker in colour than the pages that follow,
with a title and Harriot's initials. <lb/>
For a detailed account of the history and the mathematics of the treatise that follows,
see Janet Beery and Jacqueline Stedall,
<emph style="it">Thomas Harriot 19s doctrine of triangular numbers: the 'Magisteria magna'</emph> (2009).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s277" xml:space="preserve">
De Numeris Triangularibus <lb/>
et inde <lb/>
De progressionibus Arithmeticis <lb/>
Magisteria magna <lb/>
T. H.
<lb/>[<emph style="it">tr:
On triangular numbers and thence artihmetic progressions <lb/>
The great doctrine of Thomas Harriot.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f107v" o="107v" n="214"/>
<pb file="add_6782_f108" o="108" n="215"/>
<div xml:id="echoid-div69" type="page_commentary" level="2" n="69">
<p>
<s xml:id="echoid-s278" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s278" xml:space="preserve">
Throughout the treatise that follows, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> is to be read as a positive integer. <lb/>
The table at the top of the page is an array of general triangular numbers:
the first row and first column contain units, the second row and second column contain lengths,
the third row and third column contain triangular numbers, the fourth row and fourth column contain pyramidal numbers,
and so on.
Between the numbers are signs that combine 'plus' and 'equals',
illustrating the additive property of the table. <lb/>
The numerators and denominators of the fractions are to be read as products.
Thus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>7</mn><mn>8</mn><mn>9</mn><mo>,</mo><mn>1</mn><mn>0</mn><mo>,</mo><mn>1</mn><mn>1</mn></mrow><mrow><mn>1</mn><mn>2</mn><mn>3</mn><mn>4</mn><mn>5</mn></mrow></mfrac></mstyle></math> in the bottom right-hand corner, for example, is to be read as
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>7</mn><mo>×</mo><mn>8</mn><mo>×</mo><mn>9</mn><mo>×</mo><mn>1</mn><mn>0</mn><mo>×</mo><mn>1</mn><mn>1</mn></mrow><mrow><mn>1</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>4</mn><mo>×</mo><mn>5</mn></mrow></mfrac><mo>=</mo><mn>9</mn><mn>2</mn><mn>4</mn></mstyle></math>. <lb/>
Below the tables Harriot has written general formulae for the numbers in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th row.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head21" xml:space="preserve">
1.)
</head>
<pb file="add_6782_f108v" o="108v" n="216"/>
<pb file="add_6782_f109" o="109" n="217"/>
<div xml:id="echoid-div70" type="page_commentary" level="2" n="70">
<p>
<s xml:id="echoid-s280" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s280" xml:space="preserve">
The formulae from page 1 (Add MS 6782, f. 108) are expanded by long multiplication,
with each formula used as a starting point for the next.
The formula in the second box, for example, is obtained from the formula in the first box,
by multiplying by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo maxsize="1">)</mo></mstyle></math> and dividing by 3 (as also instructed by Cardano, see Add MS 6782, f. 44).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head22" xml:space="preserve">
2.)
</head>
<pb file="add_6782_f109v" o="109v" n="218"/>
<pb file="add_6782_f110" o="110" n="219"/>
<div xml:id="echoid-div71" type="page_commentary" level="2" n="71">
<p>
<s xml:id="echoid-s282" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s282" xml:space="preserve">
In the table at the top of the page, the triangular numbers from page 1 (Add MS 6782, f. 108)
are rearranged into a triangular pattern, with the sum of each row on the right. <lb/>
In the next table, each entry from the top table is written as a fraction, as on page 1. <lb/>
Below the table are general formulae for the entries in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo maxsize="1">)</mo></mstyle></math>th row.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head23" xml:space="preserve">
3.)
</head>
<pb file="add_6782_f110v" o="110v" n="220"/>
<pb file="add_6782_f111" o="111" n="221"/>
<div xml:id="echoid-div72" type="page_commentary" level="2" n="72">
<p>
<s xml:id="echoid-s284" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s284" xml:space="preserve">
The formulae from the bottom of page 3 (Add MS 6782, f. 110), expanded by long multiplication.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head24" xml:space="preserve">
4.)
</head>
<pb file="add_6782_f111v" o="111v" n="222"/>
<pb file="add_6782_f112" o="112" n="223"/>
<div xml:id="echoid-div73" type="page_commentary" level="2" n="73">
<p>
<s xml:id="echoid-s286" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s286" xml:space="preserve">
At the top of the page are two differences tables.
In each case column headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> may be taken as the starting column.
Column <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> contains successive differences between entries in column <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, and so on.
As elsewhere, a triangle broadening downwards thus, Δ, indicates an increasing column.
A small square is used to indicate columns of equal entries. <lb/>
The central table shows a difference table generated from a constant differenc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>.
Now the lower case letters <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>, <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> represent the first entry of each column
(the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, which for Harriot represented an un unknown quantity, is omitted).
Harriot has drawn a diagonal line under the table generated from a single entry
in the constant difference column.
Two small inset charts to the right of the main table show
the pattern of increasing and decreasing rows (here they are all increasing)
and the pattern of signs in each column (here they are all <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo></mstyle></math>). <lb/>
The lower table contains general formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo maxsize="1">)</mo></mstyle></math>th entry
in each column of the difference table,
using the triangular number coefficients established on page 3 (Add MS 6782, f. 110).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head25" xml:space="preserve">
5.)
</head>
<pb file="add_6782_f112v" o="112v" n="224"/>
<pb file="add_6782_f113" o="113" n="225"/>
<div xml:id="echoid-div74" type="page_commentary" level="2" n="74">
<p>
<s xml:id="echoid-s288" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s288" xml:space="preserve">
As on page 5 (Add MS 6782, f. 112), this page begins with two difference tables,
but now the columns headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> are decreasing, indicated by a triangle narrowing downwards. <lb/>
As on page 5, the central table contains formulae for the individual entries.
The two small inset charts to the right of the main table show
the pattern of increasing and decreasing rows (here they are alternately increasing and decreasing)
and the pattern of signs in each column. <lb/>
The lower table contains general formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo maxsize="1">)</mo></mstyle></math>th entry
in each column of the difference table,
using the triangular number coefficients established on page 3 (Add MS 6782, f. 110).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head26" xml:space="preserve">
6.)
</head>
<pb file="add_6782_f113v" o="113v" n="226"/>
<pb file="add_6782_f114" o="114" n="227"/>
<div xml:id="echoid-div75" type="page_commentary" level="2" n="75">
<p>
<s xml:id="echoid-s290" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s290" xml:space="preserve">
Page 7 is similar to page 6 (Add MS 6782, f. 113),
except for a change in the pattern of increasing and decreasing columns. <lb/>
The difference table at the top right contains a rare error: the last entry of column <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>
should be 1031 not 1030.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head27" xml:space="preserve">
7.)
</head>
<pb file="add_6782_f114v" o="114v" n="228"/>
<pb file="add_6782_f115" o="115" n="229"/>
<div xml:id="echoid-div76" type="page_commentary" level="2" n="76">
<p>
<s xml:id="echoid-s292" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s292" xml:space="preserve">
Here Harriot has listed all possible patterns of increasing (c) and decreasing (d) columns
for difference tables of up to six columns. <lb/>
In the lower half of the page, Harriot has produced 32 charts, like those on pages 5 to 7
(Add MS 6782, f. 111 to f. 113), showing the sign patterns in the column entries,
for each pattern of increasing and decreasing columns, for up to six columns. <lb/>
The symbols above charts 1 and 32, which look rather like <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>σ</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math>,
are Harriot's symbols for tangents and secants.
The symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> above tables 11 and 22 is his symbol for sines. In each case,
the patterns of c and d columns are those required for the corresponding trigonometric tables.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head28" xml:space="preserve">
8.)
</head>
<pb file="add_6782_f115v" o="115v" n="230"/>
<pb file="add_6782_f116" o="116" n="231"/>
<div xml:id="echoid-div77" type="page_commentary" level="2" n="77">
<p>
<s xml:id="echoid-s294" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s294" xml:space="preserve">
This page shows general entries in a difference table with six columns,
generated from 24 entries in the constant difference column.
As on page 5 (Add MS 6782, f. 112), a diagonal line has been drawn below the entries
generated from just one entry in the constant difference column.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head29" xml:space="preserve">
9.)
</head>
<pb file="add_6782_f116v" o="116v" n="232"/>
<pb file="add_6782_f117" o="117" n="233"/>
<div xml:id="echoid-div78" type="page_commentary" level="2" n="78">
<p>
<s xml:id="echoid-s296" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s296" xml:space="preserve">
The left hand side of the page contains a difference table with increasing columns,
generated from a constant difference 2. <lb/>
In the first table on the right, the column headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>
contains every third entry (denoted by the note <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>=</mo><mn>3</mn></mstyle></math>) from the column headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>. <lb/>
In the second table on the right, the column headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math>
contains every second entry (denoted by the note <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>=</mo><mn>2</mn></mstyle></math>) from the column headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. <lb/>
The other three tables on the right are constructed in a similar way.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head30" xml:space="preserve">
10.)
</head>
<pb file="add_6782_f117v" o="117v" n="234"/>
<pb file="add_6782_f118" o="118" n="235"/>
<div xml:id="echoid-div79" type="page_commentary" level="2" n="79">
<p>
<s xml:id="echoid-s298" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s298" xml:space="preserve">
This folio is very similar to the previous one (Add MS 6782, f. 117).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head31" xml:space="preserve">
11.)
</head>
<pb file="add_6782_f118v" o="118v" n="236"/>
<pb file="add_6782_f119" o="119" n="237"/>
<div xml:id="echoid-div80" type="page_commentary" level="2" n="80">
<p>
<s xml:id="echoid-s300" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s300" xml:space="preserve">
This folio is similar to the previous two (Add MS 6782, f. 117 and f. 118)
except for a different pattern of increasing and decreasing columns.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head32" xml:space="preserve">
12.)
</head>
<pb file="add_6782_f119v" o="119v" n="238"/>
<pb file="add_6782_f120" o="120" n="239"/>
<div xml:id="echoid-div81" type="page_commentary" level="2" n="81">
<p>
<s xml:id="echoid-s302" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s302" xml:space="preserve">
This folio is similar to the previous one (Add MS 6782, f. 119)
except for the opposite pattern of increasing and decreasing columns.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head33" xml:space="preserve">
13.)
</head>
<pb file="add_6782_f120v" o="120v" n="240"/>
<pb file="add_6782_f121" o="121" n="241"/>
<div xml:id="echoid-div82" type="page_commentary" level="2" n="82">
<p>
<s xml:id="echoid-s304" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s304" xml:space="preserve">
On the right hand side of the page are three partial difference tables from earlier pages;
the first and third tables are from page 10 (Add MS 6782, f. 117) while the second is from page 11 (f. 121).
In each case, Harriot has also listed the first entries of the columns of the original difference table,
denoted by <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>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. <lb/>
On the left hand side are algebraic versions of the same partial difference tables.
The first table gives formulae for successive second entries
of the column beginning with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, together with successive differences. <lb/>
The second table gives formulae for successive third entries
of the column beginning with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, together with successive differences.
The third table gives formulae for succesive <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries
of the column beginning with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. <lb/>
The final table shows the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th-entry formulae multiplied out,
with differences calculated between them.
This demonstrates that the constant difference,
in a table constructed from every <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entry of the original table, is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>n</mi><mi>n</mi><mi>a</mi></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head34" xml:space="preserve">
14.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s306" xml:space="preserve">
hoc est:
<lb/>[<emph style="it">tr:
that is:
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f121v" o="121v" n="242"/>
<pb file="add_6782_f122" o="122" n="243"/>
<div xml:id="echoid-div83" type="page_commentary" level="2" n="83">
<p>
<s xml:id="echoid-s307" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s307" xml:space="preserve">
Following from the calculations in the previous folio (Add MS 6782, f. 121)
for the column beginning with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>,
this folio gives the formulae for successive <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries
from the columns beginning with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> and with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head35" xml:space="preserve">
15.)
</head>
<pb file="add_6782_f122v" o="122v" n="244"/>
<pb file="add_6782_f123" o="123" n="245"/>
<head xml:id="echoid-head36" xml:space="preserve">
16.)
</head>
<div xml:id="echoid-div84" type="page_commentary" level="2" n="84">
<p>
<s xml:id="echoid-s309" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s309" xml:space="preserve">
Following from the calculations in the previous folios (Add MS 6782, f. 122)
for the columns beginning with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>,
this folio gives the formulae for successive <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries
from the columns beginning with <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>c</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f123v" o="123v" n="246"/>
<pb file="add_6782_f124" o="124" n="247"/>
<div xml:id="echoid-div85" type="page_commentary" level="2" n="85">
<p>
<s xml:id="echoid-s311" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s311" xml:space="preserve">
This folio shows the formulae from page 15 (Add MS 6782, f. 122) for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> column,
multiplied out in full and with differences calculated bewteen all entries.
The table is too wide to fit on the page,
so the differences appear in the boxes below the main entries.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head37" xml:space="preserve">
17.)
</head>
<pb file="add_6782_f124v" o="124v" n="248"/>
<pb file="add_6782_f125" o="125" n="249"/>
<div xml:id="echoid-div86" type="page_commentary" level="2" n="86">
<p>
<s xml:id="echoid-s313" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s313" xml:space="preserve">
This folio shows the formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> columns
from pages 15 and 16 (Add MS 6782, f. 122 and f. 123)
multiplied out in full and with differences calculated between all entries.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head38" xml:space="preserve">
18.)
</head>
<pb file="add_6782_f125v" o="125v" n="250"/>
<pb file="add_6782_f126" o="126" n="251"/>
<div xml:id="echoid-div87" type="page_commentary" level="2" n="87">
<p>
<s xml:id="echoid-s315" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s315" xml:space="preserve">
On this folio and the next (Add MS 6782, f. 126 and f. 127),
Harriot has used the formulae from page 17 (Add MS 6782, f. 124)
to write equations 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>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>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>
in terms of <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>, <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>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>.
He has solved first for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>,
then for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> in terms of <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>B</mi></mstyle></math>,
then for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> in terms of <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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>, and so on.
He has ended each calculation with 'RE', indicating 'recto' or 'correct'.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head39" xml:space="preserve">
19.)
</head>
<p>
<s xml:id="echoid-s317" xml:space="preserve">
1. Canon. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f126v" o="126v" n="252"/>
<pb file="add_6782_f127" o="127" n="253"/>
<div xml:id="echoid-div88" type="page_commentary" level="2" n="88">
<p>
<s xml:id="echoid-s318" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s318" xml:space="preserve">
This folios is the continuation of page 19 (Add MS 6782, f. 126).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head40" xml:space="preserve">
20.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s320" xml:space="preserve">
Residuum : 1. canonis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Remainder of the canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f127v" o="127v" n="254"/>
<pb file="add_6782_f128" o="128" n="255"/>
<div xml:id="echoid-div89" type="page_commentary" level="2" n="89">
<p>
<s xml:id="echoid-s321" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s321" xml:space="preserve">
On this folio Harriot has carried out calculations similar to those on pages 19 and 20
(Add MS 6782, f. 126 and f. 127),
this time to find equations 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>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>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>
in terms of <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>, <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>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math>. <lb/>
Note that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>8</mn><mi>n</mi><mi>n</mi></mstyle></math> in line 7 should be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>8</mn><mi>n</mi><mi>n</mi><mi>n</mi></mstyle></math>; the error is corrected in the next line.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head41" xml:space="preserve">
21.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s323" xml:space="preserve">
1. Canon. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f128v" o="128v" n="256"/>
<pb file="add_6782_f129" o="129" n="257"/>
<div xml:id="echoid-div90" type="page_commentary" level="2" n="90">
<p>
<s xml:id="echoid-s324" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s324" xml:space="preserve">
On this folio Harriot has carried out calculations similar to those on pages 19, 20, and 21
(Add MS 6782, f. 126, f. 127, and f. 128),
this time to find equations 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>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>
in terms of <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>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>. <lb/>
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head42" xml:space="preserve">
22.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s326" xml:space="preserve">
1. Canon. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s327" xml:space="preserve">
1. Canon. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s328" xml:space="preserve">
1. Canon. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f129v" o="129v" n="258"/>
<pb file="add_6782_f130" o="130" n="259"/>
<div xml:id="echoid-div91" type="page_commentary" level="2" n="91">
<p>
<s xml:id="echoid-s329" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s329" xml:space="preserve">
On this folio, Harriot has summarized his results from pages 19 and 20 (Add MS 6782, f. 126 and f. 127)
and has extended them to other patterns of increasing and decreasing columns. <lb/>
The symbols <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>σ</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math> in the upper tables indicate that these formulae
may be used to interpolate values in tables of tangents and secants. <lb/>
The symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> in the lower tables indicate that these formulae
may be used to interpolate tables of sines.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head43" xml:space="preserve">
23.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s331" xml:space="preserve">
Canones, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Canons for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f130v" o="130v" n="260"/>
<pb file="add_6782_f131" o="131" n="261"/>
<div xml:id="echoid-div92" type="page_commentary" level="2" n="92">
<p>
<s xml:id="echoid-s332" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s332" xml:space="preserve">
On this folio, Harriot has summarized his results from page 21 (Add MS 6782, f. 128)
and has extended them to other patterns of increasing and decreasing columns. <lb/>
The symbols <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>σ</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math> in the upper tables indicate that these formulae
may be used to interpolate values in tables of tangents and secants. <lb/>
The symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> in the lower tables indicate that these formulae
may be used to interpolate tables of sines.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head44" xml:space="preserve">
24.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s334" xml:space="preserve">
Canones, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Canons for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f131v" o="131v" n="262"/>
<pb file="add_6782_f132" o="132" n="263"/>
<div xml:id="echoid-div93" type="page_commentary" level="2" n="93">
<p>
<s xml:id="echoid-s335" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s335" xml:space="preserve">
On this folio, Harriot has summarized his results from page 22 (Add MS 6782, f. 129)
and has extended them to other patterns of increasing and decreasing columns. <lb/>
The symbols <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>σ</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math> in the upper tables indicate that these formulae
may be used to interpolate tables of tangents and secants. <lb/>
The symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> in the lower tables indicate that these formulae
may be used to interpolate tables of sines.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head45" xml:space="preserve">
25.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s337" xml:space="preserve">
Canones, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Canons for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s338" xml:space="preserve">
Canones, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Canons for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f132v" o="132v" n="264"/>
<pb file="add_6782_f133" o="133" n="265"/>
<div xml:id="echoid-div94" type="page_commentary" level="2" n="94">
<p>
<s xml:id="echoid-s339" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s339" xml:space="preserve">
This folio contains a set of formulae for interpolating <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo maxsize="1">)</mo></mstyle></math> new terms between each pair of entries
in the fourth column of a difference table (the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> column).
For reasons of space, Harriot has written the entries for the fourth column below those
for the first, second, and third columns. <lb/>
At the bottom of the page, he has written a single general formula for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math>th interpolated term
in the fourth column. This he calls the 'magisterium', which may here be translated as 'rule'.
Since this formula is expressed entirely in terms of <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><mi>N</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>C</mi></mstyle></math>, <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>,
he longer needs to compute <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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head46" xml:space="preserve">
26.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s341" xml:space="preserve">
Pro Magisterio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
For the rule for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s342" xml:space="preserve">
Magisterium.
<lb/>[<emph style="it">tr:
Rule
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f133v" o="133v" n="266"/>
<pb file="add_6782_f134" o="134" n="267"/>
<div xml:id="echoid-div95" type="page_commentary" level="2" n="95">
<p>
<s xml:id="echoid-s343" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s343" xml:space="preserve">
This page shows formulae analogous to the 'magisterium' on page 26 (Add MS 6782, f. 133)
for the interpolation of difference tables of up to six columns, with all columns increasing. <lb/>
The symbols <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>σ</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math> indicate that these formulae
may be used to interpolate tables of tangents and secants. <lb/>
The symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math> in this context is not a fraction,
but indicates that the expression to the left of it
is the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math>th entry of a table interpolated to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> times its original length.
The sign )=( may therefore be read as 'indexed by'.
On BL Add MS 6787, f. 352, Harriot experiments with various alternatives for the symbol )=(.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head47" xml:space="preserve">
27.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s345" xml:space="preserve">
Magisteria
<lb/>[<emph style="it">tr:
Rules
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f134v" o="134v" n="268"/>
<pb file="add_6782_f135" o="135" n="269"/>
<head xml:id="echoid-head48" xml:space="preserve">
28.)
</head>
<div xml:id="echoid-div96" type="page_commentary" level="2" n="96">
<p>
<s xml:id="echoid-s346" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s346" xml:space="preserve">
Interpolation formulae as on page 27 (Add MS 6782, f. 134) but now with all columns decreasing. <lb/>
The symbols <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>σ</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math> indicate that these formulae
may be used to interpolate tables of tangents and secants.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s348" xml:space="preserve">
Magisteria
<lb/>[<emph style="it">tr:
Rules
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f135v" o="135v" n="270"/>
<pb file="add_6782_f136" o="136" n="271"/>
<div xml:id="echoid-div97" type="page_commentary" level="2" n="97">
<p>
<s xml:id="echoid-s349" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s349" xml:space="preserve">
Interpolation formulae as on page 27 (Add MS 6782, f. 134) but now
for columns that are alternately increasing and decreasing. <lb/>
The symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> indicates that these formulae may be used to interpolate tables of sines.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head49" xml:space="preserve">
29.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s351" xml:space="preserve">
Magisteria
<lb/>[<emph style="it">tr:
Rules
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f136v" o="136v" n="272"/>
<pb file="add_6782_f137" o="137" n="273"/>
<div xml:id="echoid-div98" type="page_commentary" level="2" n="98">
<p>
<s xml:id="echoid-s352" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s352" xml:space="preserve">
Interpolation formulae as on page 29 (Add MS 6782, f. 136) but now
for columns that are alternately decreasing and increasing. <lb/>
The symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> indicates that these formulae may be used to interpolate tables of sines.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head50" xml:space="preserve">
30.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s354" xml:space="preserve">
Magisteria
<lb/>[<emph style="it">tr:
Rules
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f137v" o="137v" n="274"/>
<pb file="add_6782_f138" o="138" n="275"/>
<div xml:id="echoid-div99" type="page_commentary" level="2" n="99">
<p>
<s xml:id="echoid-s355" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s355" xml:space="preserve">
The numbers 4, 3, 2, and 1 in the upper righthand corners of pages 31, 32, 33, and 34,
(Add MS 6782, f. 138 to f. 141) indicate that these four folios are closely related
and could be read in either direction. <lb/>
This folio brings together the formulae from pages 27 to 30 (Add MS 6782, f. 134 to f. 137). <lb/>
The coefficients of <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>, <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>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>
are now written as descending rather than ascending powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math>.
This means that the signs of the coefficients now follow the patterns given in the sign charts
for all increasing, or all decreasing, or alternately increasing and decreasing columns.<lb/>
The symbols <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>σ</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math> below the upper table indicate that these patterns
are required for interpolating tables of tangents and secants. <lb/>
The symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> below the lower table indicates that these patterns
are required for interpolating tables of sines.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head51" xml:space="preserve">
31.)
</head>
<p>
<s xml:id="echoid-s357" xml:space="preserve">
(4.
</s>
</p>
<pb file="add_6782_f138v" o="138v" n="276"/>
<pb file="add_6782_f139" o="139" n="277"/>
<div xml:id="echoid-div100" type="page_commentary" level="2" n="100">
<p>
<s xml:id="echoid-s358" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s358" xml:space="preserve">
This folio contains the same interpolation formulae as on page 31,
but now the coefficients of <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>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> have been factorized. <lb/>
The inclusion of columns headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> in the inset charts, on this and the two following pages
(Add MS 6782, f. 140 and f. 141), emphasizes that the formulae may be generalized
to any number of columns. <lb/>
The symbols <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>σ</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math> below the lower left table indicate that these patterns
are required for interpolating tables of tangents and secants. <lb/>
The symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> below the lower right table indicates that these patterns
are required for interpolating tables of sines.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head52" xml:space="preserve">
32.)
</head>
<p>
<s xml:id="echoid-s360" xml:space="preserve">
(3.
</s>
</p>
<pb file="add_6782_f139v" o="139v" n="278"/>
<pb file="add_6782_f140" o="140" n="279"/>
<div xml:id="echoid-div101" type="page_commentary" level="2" n="101">
<p>
<s xml:id="echoid-s361" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s361" xml:space="preserve">
This folio contains another version of the interpolation formulae on pages 31 and 32
(Add MS 6782, f. 138 and f. 139),
but now each coefficient of <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>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> has its own denominator.
This appears to be Harriot's preferred form. <lb/>
The symbols <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>σ</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math> below the lower left table indicate that these patterns
are required for interpolating tables of tangents and secants. <lb/>
The symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> below the lower right table indicates that these patterns
are required for interpolating tables of sines.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head53" xml:space="preserve">
33.)
</head>
<p>
<s xml:id="echoid-s363" xml:space="preserve">
(2.
</s>
</p>
<pb file="add_6782_f140v" o="140v" n="280"/>
<pb file="add_6782_f141" o="141" n="281"/>
<div xml:id="echoid-div102" type="page_commentary" level="2" n="102">
<p>
<s xml:id="echoid-s364" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s364" xml:space="preserve">
This folio contains the same formulae as page 33 (Add MS 6782, f. 140)
but with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math> now replaced by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math>.
That is, there is no interpolation. <lb/>
This page is numbered 1 in its upper righthand corner,
making it the first in the subsequence of pages 34, 33, 32, and 31 (Add MS f. 141 to f. 138).
Harriot explained on the next page (Add MS 6782, f. 142), which is a second version of this one,
also numbered 1 in its upper righthand corner,
that one may begin with the formulae on page 34 (Add MS 6782, 141),
then derive the formulae on page 33 (Add MS 6782, f. 140) by replacing <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math> by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math>.
That is, one can reverse the sequence of pages 31 to 34. <lb/>
As on pages 32 and 33 (Add MS 6782, f. 139 and f. 140)
the sign charts show how to adapt the formulae to different patterns
of increasing and decreasing columns. <lb/>
The symbols <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>σ</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math> below the lower left table indicate that these patterns
are required for interpolating tables of tangents and secants. <lb/>
The symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> below the lower right table indicates that these patterns
are required for interpolating tables of sines.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head54" xml:space="preserve">
34.)
</head>
<p>
<s xml:id="echoid-s366" xml:space="preserve">
(1.
</s>
</p>
<pb file="add_6782_f141v" o="141v" n="282"/>
<pb file="add_6782_f142" o="142" n="283"/>
<div xml:id="echoid-div103" type="page_commentary" level="2" n="103">
<p>
<s xml:id="echoid-s367" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s367" xml:space="preserve">
On this folio, Harriot explains the relationship between the formulae on pages 33 and 34
(Add MS 6782, f. 140 and f. 141), and their common origin in those that appeared earlier,
on page 5 (Add MS 6782, f. 112). <lb/>
In the lower half of the page, just below the dividing line,
Harriot replaces <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math> by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math> in formula 3) from page 34,
arriving at formula 3) from page 33.
In this case, he is using <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math> to denote an ordinary fraction.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head55" xml:space="preserve">
34.) 2<emph style="super">o</emph>.)
</head>
<p>
<s xml:id="echoid-s369" xml:space="preserve">
1.)
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s370" xml:space="preserve">
Etsi species quæ habentur (pag: 34.1.) ortum ducunt ex (pag: 33.2.) <lb/>
Attamen primam originem videre licet pag. 5. ubi illæ omnes <lb/>
appareat notatæ.
<lb/>[<emph style="it">tr:
Although the cases we have on page 34.1 arise from those on page 33.2,
nevertheless one may see their origins on page 5 where all of them appear in notation.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s371" xml:space="preserve">
Utile etiam ac incundum est, considerare harum reductionum (vide versa) <lb/>
ad species in pag: 33.2. quæ huius operis sunt magisteria maxima.
<lb/>[<emph style="it">tr:
It is useful and also pleasing to consider (conversely) the reduction to the cases on page 33.2,
which are the most important rules of this work.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s372" xml:space="preserve">
Examplum unum sufficiet.
<lb/>[<emph style="it">tr:
One example will suffice.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s373" xml:space="preserve">
Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi><mo>=</mo><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math>.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi><mo>=</mo><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math>
</emph>]<lb/>
</s>
<s xml:id="echoid-s374" xml:space="preserve">
Et species reducta erit: (ut pag: 33.2.) et ut sequitur:
<lb/>[<emph style="it">tr:
And the cases will be reduced (as on page 33.2) and as follows:
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s375" xml:space="preserve">
Fit ita: Si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi><mo>=</mo><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math> <lb/>
erit:
<lb/>[<emph style="it">tr:
Let it be done thus: <lb/>
then:
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s376" xml:space="preserve">
Et sic de alijs speciebus.
<lb/>[<emph style="it">tr:
And so on for other cases.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f142v" o="142v" n="284"/>
<pb file="add_6782_f143" o="143" n="285"/>
<div xml:id="echoid-div104" type="page_commentary" level="2" n="104">
<p>
<s xml:id="echoid-s377" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s377" xml:space="preserve">
On this and the following folio (Add MS 6782, f. 144),
Harriot gives numerical examples of his interpolation method. <lb/>
At the top of the page are formulae for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math> and for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math>,
for difference tables with two columns.
Below that are four examples of tables with two columns. <lb/>
The table on the left, with columns headed <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><mi>B</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>, is interpolated
first to six, then four, then five times the number of original entries;
that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> takes the values 6, then 4, then 5.
The symbol * next to the interpolated tables marks entries from the original lefthand table. <lb/>
Below the tables, the first column of working uses the formula for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math> from the top of the page,
to obtain the entries 17, 77, and 149 in the difference table on the left. <lb/>
The second column of working uses the formula for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math> from the top of the page,
to obtain one entry in each of the three remaining difference tables. <lb/>
The third column of working presents the converse problem,
showing how to solve for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math> ),
given values of <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>B</mi></mstyle></math> and an entry <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>.
Where the solution is not an integer, Harriot replaces <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math> by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head56" xml:space="preserve">
35.)
</head>
<pb file="add_6782_f143v" o="143v" n="286"/>
<pb file="add_6782_f144" o="144" n="287"/>
<div xml:id="echoid-div105" type="page_commentary" level="2" n="105">
<p>
<s xml:id="echoid-s379" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s379" xml:space="preserve">
This folio gives further numerical examples of interpolation for tables with three columns. <lb/>
As on page 35 (Add MS 6782, f. 143), Harriot gives the appropriate <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math> formulae
at the top of the page. The formula in the first line is from page 33 (Add MS 6782, f. 140),
while the formulae on the second line are from page 32 (Add MS 6782, f. 139). <lb/>
The difference table on the left (headed <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><mi>C</mi></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>A</mi></mstyle></math>) is interpolated
to five times, and then twice, the number of original entries;
that is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> takes the values 5 and 2, respectively. <lb/>
The working in the first column illustrates the use of the formula for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math>
to calculate values in the left hand table. <lb/>
The working in the second column illustrates the use of the formula for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math>
to calculate the first new value in the second table (indexed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>5</mn></mrow></mfrac></mstyle></math>),
and the third new value in the third table (indexed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>). <lb/>
The working in the third column shows, as on page 35 (Add MS 6782, f. 143), how to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math>
given values of <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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>, and an entry <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>.
In this case the working leads to a quadratic equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math>,
which Harriot solves by completing the square.
As on page 35, the solution is not an integer, and so Harriot replaces <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math> by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head57" xml:space="preserve">
36.)
</head>
<pb file="add_6782_f144v" o="144v" n="288"/>
<pb file="add_6782_f145" o="145" n="289"/>
<div xml:id="echoid-div106" type="page_commentary" level="2" n="106">
<p>
<s xml:id="echoid-s381" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s381" xml:space="preserve">
This folios contains an interpolation formula from the 'Magisteria'
but instead of <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>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, ...
we now have <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><mrow><msup><mi>p</mi><mn>1</mn></msup></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>2</mn></msup></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>3</mn></msup></mrow></mstyle></math>, ... (these are superscripts, not powers). <lb/>
The notation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>P</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>D</mi><mn>1</mn></msup></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>D</mi><mn>2</mn></msup></mrow></mstyle></math>, ... at the top of the page is unusual;
the formula is otherwise identical to formula 5) from page 33.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f145v" o="145v" n="290"/>
<pb file="add_6782_f146" o="146" n="291"/>
<div xml:id="echoid-div107" type="page_commentary" level="2" n="107">
<p>
<s xml:id="echoid-s383" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s383" xml:space="preserve">
Interpolated tables, see page 35 of the 'Magisteria' (Add MS 6782, f. 143).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f146v" o="146v" n="292"/>
<div xml:id="echoid-div108" type="page_commentary" level="2" n="108">
<p>
<s xml:id="echoid-s385" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s385" xml:space="preserve">
This folio contains interpolation formulae similar to those on page 32 (Add MS 6782, f. 139). <lb/>
It also contains a title similar, but not identical, to that at the beginning of the 'Magisteria'
(Add MS 6782, f. 107).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s387" xml:space="preserve">
THOMÆ HARIOTI <lb/>
Magisteria <lb/>
Numerorum Trangularium <lb/>
et inde <lb/>
Progressionum Arithmeticarum <lb/>
(veteribus et recentioribus ignota) <lb/>
incognita)
<lb/>[<emph style="it">tr:
THOMAS HARRIOT'S doctrine of triangular numbers and thence arithmetic progressions <lb/>
(unkonwn and unrecognized by ancient and more recent authors)
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f147" o="147" n="293"/>
<div xml:id="echoid-div109" type="page_commentary" level="2" n="109">
<p>
<s xml:id="echoid-s388" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s388" xml:space="preserve">
The two lines at the end of the page have nothing to do with the mathematics above them,
and are not apparently connected with each other.
<foreign xml:lang="lat">'Porcus per taurum sequitur vestigia ferri'</foreign>
is the first line of an epitaph engraved on the tombstone of Edmund Bunny,
rector of Bolton Percy and canon of York,
who died in February 1617/8 and was interred in York Minster:
</s>
<lb/>
<quote xml:lang="lat">
Porcus per taurum sequitur vestigia ferri Â <lb/>
Anser ovem maculat, cui potum vacca ministrat. <lb/>
Expone et redde sensum
</quote>
<lb/>
<s xml:id="echoid-s389" xml:space="preserve">
This is an illustration of synecdoche, in which a part is referred to as the whole;
thus the pig (lard, here as used to lubricate cobbler's thread) follows the footprints of the iron (needle)
through the bull (leather); the goose (quill), to whom the cow (inkhorn) provides drink (ink), stains the sheep (skin).
(The standard version has
<foreign xml:lang="lat">variat</foreign> instead of <foreign xml:lang="lat">maculat</foreign>.)
<foreign xml:lang="lat">Expone et redde sensum</foreign>
is an instruction to the student: 'explain and translate'.
It ispossible that <foreign xml:lang="lat">porcus</foreign> is a play on Percy.
If the York epitaph was Harriot's source, it gives us a possible date of 1618 for this folio.
</s>
<lb/>
<s xml:id="echoid-s390" xml:space="preserve">
'Bombardagladiofunhastiflammiloquentes'
(Breathing bombs, swords, death, spears, and flames)
is from a Latin translation by the 16th-century German humanist Martin Crucius of a Greek verse
(perhaps also by Crucius) consisting of compound words.
The verse appears in the preface to the <emph style="it">Opera omnnia theologica</emph> (1583) of Lambert Daneau
and in the third edition of his <emph style="it">Elenchi hæreticorum</emph> (1592).
The same line appears also in Add MS 6788, f. 50.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head58" xml:space="preserve" xml:lang="lat">
A.1. Ad numeros triangulos <lb/>
quadratos <lb/>
pentagonos &amp;c. <lb/>
et illorum progenies
<lb/>[<emph style="it">tr:
On triangular, square, pentagonal numbers, and their progeny
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s392" xml:space="preserve">
Generaliter
<lb/>[<emph style="it">tr:
Generally
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s393" xml:space="preserve">
Ad progressiones arithmeticas <lb/>
incipientes ab unitate vel quovis <lb/>
numero; quolibet etiam excessu <lb/>
progredientes.
<lb/>[<emph style="it">tr:
For arithmetic porgressions starting from one or any number; proceeding with whatever excess.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s394" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. primus numerus in progressione. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. excessus. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>. numerus loci in progressione.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. the first number in the progression <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. the excess <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>. the number of places in the progression
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s395" xml:space="preserve">
Quomodo istæ æqautiones <lb/>
continuentur ad infinitum <lb/>
apparet in altera charta.
<lb/>[<emph style="it">tr:
By what meanst these equations may be continued indefinitely appears in another sheet.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s396" xml:space="preserve">
porcus per taurum sequitur vestigia ferri. <lb/>
Bombardagladiofunhastiflammiloquentes.
<lb/>[<emph style="it">tr:
The pig follows the footsteps of the iron through the bull. <lb/>
Breathing bombs, swords, death, spears, and flames.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f147v" o="147v" n="294"/>
<pb file="add_6782_f148" o="148" n="295"/>
<div xml:id="echoid-div110" type="page_commentary" level="2" n="110">
<p>
<s xml:id="echoid-s397" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s397" xml:space="preserve">
Rules for the numbers in six successive columns,
generated from a constant difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> in column 0.
The first entry in each column is also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. <lb/>
The expression in the first box, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi></mstyle></math>,
gives the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entry in the first column (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mi>p</mi></mstyle></math>, and so on). <lb/>
The expression in the second box, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi><mi>n</mi><mo>+</mo><mn>1</mn><mi>p</mi><mi>n</mi></mstyle></math>, divided by 2,
gives the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entry in the second column (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>0</mn><mi>p</mi></mstyle></math>, and so on). <lb/>
The expression in the third box, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi><mi>n</mi><mi>n</mi><mo>+</mo><mn>3</mn><mi>p</mi><mi>n</mi><mi>n</mi><mo>+</mo><mn>2</mn><mi>p</mi><mi>n</mi></mstyle></math>, divided by 6,
gives the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entry in the third column (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>0</mn><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>0</mn><mi>p</mi></mstyle></math>, and so on). <lb/>
The layout shows how the general term in row <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mo>+</mo><mn>1</mn></mstyle></math>
is generated from the general term in row <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math>,
by multiplying by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>+</mo><mi>k</mi></mstyle></math> and dividing by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mo>+</mo><mn>1</mn></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head59" xml:space="preserve" xml:lang="lat">
A.1.) Ad numeros triangulos et illorum progenies.
<lb/>[<emph style="it">tr:
On triangular numbers and their progeny
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s399" xml:space="preserve">
In Via Generali.
<lb/>[<emph style="it">tr:
In a general way
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f148v" o="148v" n="296"/>
<pb file="add_6782_f149" o="149" n="297"/>
<div xml:id="echoid-div111" type="page_commentary" level="2" n="111">
<p>
<s xml:id="echoid-s400" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s400" xml:space="preserve">
This folio shows how to generate the coeffcients from the previous page, f. 148.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head60" xml:space="preserve" xml:lang="lat">
A.2. Ad æquationes numerorum figuratorum <lb/>
ut continuentur ad libitum.
<lb/>[<emph style="it">tr:
On equations of figurate numbers so that they may be continued at will
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s402" xml:space="preserve">
In genere.
<lb/>[<emph style="it">tr:
In general
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f149v" o="149v" n="298"/>
<pb file="add_6782_f150" o="150" n="299"/>
<div xml:id="echoid-div112" type="page_commentary" level="2" n="112">
<p>
<s xml:id="echoid-s403" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s403" xml:space="preserve">
This folio shows calculations similar to those on f. 148, but now <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>+</mo><mi>k</mi></mstyle></math>
has been replaced by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mo>+</mo><mi>k</mi><mi>d</mi></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head61" xml:space="preserve" xml:lang="lat">
A.3. Ad æquationes numerorum figuratorum <lb/>
ut continuentur ad libitum.
<lb/>[<emph style="it">tr:
On equations of figurate numbers so that they may be continued at will
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s405" xml:space="preserve">
in Genere.
<lb/>[<emph style="it">tr:
in general
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f150v" o="150v" n="300"/>
<pb file="add_6782_f151" o="151" n="301"/>
<pb file="add_6782_f151v" o="151v" n="302"/>
<pb file="add_6782_f152" o="152" n="303"/>
<pb file="add_6782_f152v" o="152v" n="304"/>
<pb file="add_6782_f153" o="153" n="305"/>
<div xml:id="echoid-div113" type="page_commentary" level="2" n="113">
<p>
<s xml:id="echoid-s406" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s406">
In the upper left table, the key column is the one beneath the sketch of a triangular prism.
The numbers beneath the sketch are those needed to construct triangular prisms
with length equal to one side of the base: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>=</mo><mn>1</mn><mo>×</mo><mn>1</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mo>=</mo><mn>2</mn><mo>×</mo><mn>3</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>8</mn><mo>=</mo><mn>3</mn><mo>×</mo><mn>6</mn></mstyle></math>,
and so on.
These are also the pentagonal-pyramidal numbers 1, 6, 18, ... as calculated in f. 156. <lb/>
Successive sums of these numbers are shown in the column to the left headed S (for sum). <lb/>
Successive differences are shown in the columns to the right.
</s>
<lb/>
<s xml:id="echoid-s407">
In the upper right table, the key column is the one beneath the sketch of a cube.
The numbers beneath the sketch are the cube numbers, which can also be thought of
(in keeping with the previous table) as the numbers needed to construct square prisms
with height equal to one side of the base: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>=</mo><mn>1</mn><mo>×</mo><mn>1</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>8</mn><mo>=</mo><mn>2</mn><mo>×</mo><mn>4</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>7</mn><mo>=</mo><mn>3</mn><mo>×</mo><mn>9</mn></mstyle></math>,
and so on. <lb/>
Successive sums are shown in the column to the left headed S (for sum). <lb/>
Successive differences are shown in the columns to the right. <lb/>
The column beginning 1, 7, 19, 37, ... is marked 'hexagonae equianguli 19 (equiangled hexagons);
it is possible that at first glance Harriot mistook this column for
the the hexagonal-pyramidal numbers 1, 7, 22, 50, ....
</s>
<lb/>
<s xml:id="echoid-s408">
In the lower left table, the key column is the one beneath the sketch of a pentagonal prism.
The numbers beneath the sketch are the numbers needed to construct pentagonal prisms
with length equal to one side of the base: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>=</mo><mn>1</mn><mo>×</mo><mn>1</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>0</mn><mo>=</mo><mn>2</mn><mo>×</mo><mn>5</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mn>6</mn><mo>=</mo><mn>3</mn><mo>×</mo><mn>1</mn><mn>2</mn></mstyle></math>,
and so on. <lb/>
Successive sums are shown in the column to the left headed S (for sum). <lb/>
Successive differences are shown in the columns to the right.
</s>
<lb/>
<s xml:id="echoid-s409">
In the lower right table, the key column is the one beneath the sketch of a hexagonal prism.
The numbers beneath the sketch are the numbers needed to construct hexagonal prisms
with length equal to one side of the base: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>=</mo><mn>1</mn><mo>×</mo><mn>1</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>2</mn><mo>=</mo><mn>2</mn><mo>×</mo><mn>6</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mn>5</mn><mo>=</mo><mn>3</mn><mo>×</mo><mn>1</mn><mn>5</mn></mstyle></math>,
and so on. <lb/>
Successive sums are shown in the column headed to the left headed S (for sum). <lb/>
Successive differences are shown in the columns to the right.
</s>
<lb/>
<s xml:id="echoid-s410" xml:space="preserve">
The formulae at the bottom of the page are for sums of
triangular, square, pentagonal, and hexagonal prisms.
Thus the sum of the first two triangular prisms is shown in the upper left table as 1 + 6 = 7.
The same number may be calculated from the formula <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>3</mn><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>0</mn><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>9</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>2</mn><mi>a</mi></mrow><mrow><mn>2</mn><mn>4</mn></mrow></mfrac></mstyle></math>
by putting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn></mstyle></math>. <lb/>
For clues as to how Harriot found and tested these formulae,
see ARITHMETIC/Figurate numbers/Fitting polynomials to sequences.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s412" xml:space="preserve">
columnæ [triangulari]æ. <lb/>
quæ sint pyramides [pentagones].
<lb/>[<emph style="it">tr:
Triangular prism numbers, which are also pentagonal pyramidals
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s413" xml:space="preserve">
columnæ tetragonæ <lb/>
seu cubi.
<lb/>[<emph style="it">tr:
Four-sided prism numbers, or cubes
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s414" xml:space="preserve">
Hexagones æquianguli.
<lb/>[<emph style="it">tr:
Equiangular hexagonals
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s415" xml:space="preserve">
columnæ [pentagon]æ.
<lb/>[<emph style="it">tr:
Pentagonal prisms
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s416" xml:space="preserve">
columnæ <reg norm="hexagonae" type="abbr">Hexag</reg>.
<lb/>[<emph style="it">tr:
hexagonal prisms
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s417" xml:space="preserve">
per Reductionem
<lb/>[<emph style="it">tr:
by reduction
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f153v" o="153v" n="306"/>
<pb file="add_6782_f154" o="154" n="307"/>
<div xml:id="echoid-div114" type="page_commentary" level="2" n="114">
<p>
<s xml:id="echoid-s418" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s418" xml:space="preserve">
The numerical tables on the right list, from right to left:
units, lengths, triangular numbers, triangular-pyramidal naumbers, and so on. <lb/>
The expressions on the left are general formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>th entry in each column.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head62" xml:space="preserve" xml:lang="lat">
De numeris triangulis
<lb/>[<emph style="it">tr:
On triangular numbers
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s420" xml:space="preserve">
(<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>) est numerus locorum <lb/>
seu radix.
<lb/>[<emph style="it">tr:
(<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>) is the number of places, or the root.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s421" xml:space="preserve">
Quomodo istæ æquationes <lb/>
continuentur ad libitum <lb/>
apparet in Chartis. A.
<lb/>[<emph style="it">tr:
How these equations may be continued at will is shown in sheet A.
</emph>]<lb/>
<sc>
Sheet A is probably Add MS 6782, f. 237.
</sc>
</s>
</p>
<p>
<s xml:id="echoid-s422" xml:space="preserve">
The difference of difference of <lb/>
squares is 2. <lb/>
The <reg norm="difference" type="abbr">diff</reg>.
of <reg norm="difference" type="abbr">diff</reg>.
of <reg norm="difference" type="abbr">diff</reg>: <lb/>
of cubes is 6 <lb/>
of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>: is 24. <lb/>
of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> is 12.
</s>
</p>
<p>
<s xml:id="echoid-s423" xml:space="preserve">
Of progressions the same <lb/>
make <lb/>
Triangular <lb/>
Square <lb/>
Pentagonall <lb/>
nombers
&amp;c.
</s>
</p>
<pb file="add_6782_f154v" o="154v" n="308"/>
<pb file="add_6782_f155" o="155" n="309"/>
<div xml:id="echoid-div115" type="page_commentary" level="2" n="115">
<p>
<s xml:id="echoid-s424" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s424" xml:space="preserve">
The numerical table at the top of the page lists, from right to left:
odd numbers, square numbers, square-pyramidal numbers, and so on. <lb/>
The column on the far right is headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> for root, that is,
the length of the side of each figure. <lb/>
An intercalated column shows a constant difference 2. <lb/>
The column under the short line (the side) gives the sequence 1, 3, 5, ...,
in which the differences are 2. <lb/>
The column under the square gives the square numbers 1, 4, 9, ...,
whose differences are 1, 3, 5, .... <lb/>
The column under the square-pyramid gives the square-pyramidal numbers 1, 5, 14, ...,
whose differences are 1, 4, 9, .... <lb/>
And so on. <lb/>
The expressions below the table are general formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>th entry in each column.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f155v" o="155v" n="310"/>
<pb file="add_6782_f156" o="156" n="311"/>
<div xml:id="echoid-div116" type="page_commentary" level="2" n="116">
<p>
<s xml:id="echoid-s426" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s426" xml:space="preserve">
The numerical table at the top of the page lists
pentagonal numbers, pentagonal-pyramidal numbers, and so on. <lb/>
The column on the far right is headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> for root, that is,
the length of the side of each figure. <lb/>
An intercalated column shows a constant difference 3. <lb/>
The column under the short line (the side) gives the sequence 1, 4, 7, ...,
in which the differences are 3. <lb/>
The column under the pentagon gives the pentagonal numbers 1, 5, 12, ...,
whose differences are 1, 4, 7, .... <lb/>
The column under the pentagonal-pyramid gives the pentagonal-pyramidal numbers 1, 6, 18, ...,
whose differences are 1, 5, 12, .... <lb/>
And so on. <lb/>
The expressions below the table are general formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>th entry in each column.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s428" xml:space="preserve">
Of the rootes.
</s>
</p>
<p>
<s xml:id="echoid-s429" xml:space="preserve">
A unit is the first [first pentagonal number].
</s>
</p>
<p>
<s xml:id="echoid-s430" xml:space="preserve">
Omitte the first &amp; the summe <lb/>
of the next two, is 5. [the second pentagonal number]
</s>
</p>
<p>
<s xml:id="echoid-s431" xml:space="preserve">
Omitte two rootes, &amp; the summe <lb/>
of the next 3, is 12. [the third pentagonal number]
</s>
</p>
<p>
<s xml:id="echoid-s432" xml:space="preserve">
Omitte 3 rootes, &amp; the summe <lb/>
of the next 4, is 22. [the fourth pentagonal number] <lb/>
&amp;c.
</s>
</p>
<pb file="add_6782_f156v" o="156v" n="312"/>
<pb file="add_6782_f157" o="157" n="313"/>
<div xml:id="echoid-div117" type="page_commentary" level="2" n="117">
<p>
<s xml:id="echoid-s433" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s433" xml:space="preserve">
The numerical table at the top of the page lists
hexagonal numbers, hexagonal-pyramidal numbers, and so on. <lb/>
The column on the far right is headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> for root, that is,
the length of the side of each figure. <lb/>
An intercalated column shows a constant difference 4. <lb/>
The column under the short line (the side) gives the sequence 1, 5, 9, ...,
in which the differences are 4. <lb/>
The column under the hexagon gives the hexagonal numbers 1, 6, 15, ...,
whose differences are 1, 5, 9, .... <lb/>
The column under the hexagonal-pyramid gives the hexagonal-pyramidal numbers 1, 7, 22, ...,
whose differences are 1, 6, 15, .... <lb/>
And so on. <lb/>
The expressions below the table are general formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>th entry in each column.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f157v" o="157v" n="314"/>
<pb file="add_6782_f158" o="158" n="315"/>
<div xml:id="echoid-div118" type="page_commentary" level="2" n="118">
<p>
<s xml:id="echoid-s435" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s435" xml:space="preserve">
The numerical table at the top of the page lists
heptagonal numbers, heptagonal-pyramidal numbers, and so on. <lb/>
The column on the far right is headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> for root, that is,
the length of the side of each figure. <lb/>
An intercalated column shows a constant difference 5. <lb/>
The column under the short line (the side) gives the sequence 1, 6, 11, ...,
in which the differences are 5. <lb/>
The column under the heptagon gives the heptagonal numbers 1, 7, 18, ...,
whose differences are 1, 6, 11, .... <lb/>
The column under the heptagonal-pyramid gives the heptagonal-pyramidal numbers 1, 8, 26, ...,
whose differences are 1, 7, 18, .... <lb/>
And so on. <lb/>
The expressions below the table are general formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>th entry in each column.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f158v" o="158v" n="316"/>
<pb file="add_6782_f159" o="159" n="317"/>
<div xml:id="echoid-div119" type="page_commentary" level="2" n="119">
<p>
<s xml:id="echoid-s437" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s437" xml:space="preserve">
The numerical table at the top of the page lists
octagonal numbers, octagonal-pyramidal numbers, and so on. <lb/>
The column on the far right is headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> for root, that is,
the length of the side of each figure. <lb/>
An intercalated column shows a constant difference 6. <lb/>
The column under the short line (the side) gives the sequence 1, 7, 13, ...,
in which the differences are 6. <lb/>
The column under the octagon gives the octagonal numbers 1, 8, 21, ...,
whose differences are 1, 7, 13, .... <lb/>
The column under the octagonal-pyramid gives the octagonal-pyramidal numbers 1, 9, 30, ...,
whose differences are 1, 8, 21, .... <lb/>
And so on. <lb/>
The expressions below the table are general formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>th entry in each column.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f159v" o="159v" n="318"/>
<pb file="add_6782_f160" o="160" n="319"/>
<pb file="add_6782_f160v" o="160v" n="320"/>
<pb file="add_6782_f161" o="161" n="321"/>
<pb file="add_6782_f161v" o="161v" n="322"/>
<pb file="add_6782_f162" o="162" n="323"/>
<pb file="add_6782_f162v" o="162v" n="324"/>
<pb file="add_6782_f163" o="163" n="325"/>
<div xml:id="echoid-div120" type="page_commentary" level="2" n="120">
<p>
<s xml:id="echoid-s439" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s439" xml:space="preserve">
Here as on Add MS 6782, f. 330, Harriot is using <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>v</mi></mstyle></math> notation, with superscripts 0, 1, 2, 3, ...
for successive entries in a general row of the table.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head63" xml:space="preserve" xml:lang="lat">
Elementa triangularium.
<lb/>[<emph style="it">tr:
The elements of triangular numbers
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s441" xml:space="preserve">
Rationes compositas: et <lb/>
componentes.
<lb/>[<emph style="it">tr:
Ratios compounded, and their components.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f163v" o="163v" n="326"/>
<pb file="add_6782_f164" o="164" n="327"/>
<div xml:id="echoid-div121" type="page_commentary" level="2" n="121">
<p>
<s xml:id="echoid-s442" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s442" xml:space="preserve">
A table that appears to have been produced from the various rules set out on f. 163.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f164v" o="164v" n="328"/>
<pb file="add_6782_f165" o="165" n="329"/>
<div xml:id="echoid-div122" type="page_commentary" level="2" n="122">
<p>
<s xml:id="echoid-s444" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s444" xml:space="preserve">
Some experiments with Pascal's triangle. <lb/>
On the left, and again at the bottom of the page, the tables have been extended upwards,
giving rise to negative triangular numbers. <lb/>
On the right, the table has been multiplied throughout by 3; see also Add MS 6785, f. 83. <lb/>
Note Harriot's use of superscript notation (not powers): <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>2</mn></msup></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>3</mn></msup></mrow></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s446" xml:space="preserve">
Male
<lb/>[<emph style="it">tr:
badly done
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s447" xml:space="preserve">
Bene
<lb/>[<emph style="it">tr:
better
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f165v" o="165v" n="330"/>
<pb file="add_6782_f166" o="166" n="331"/>
<pb file="add_6782_f166v" o="166v" n="332"/>
<pb file="add_6782_f167" o="167" n="333"/>
<pb file="add_6782_f167v" o="167v" n="334"/>
<pb file="add_6782_f168" o="168" n="335"/>
<pb file="add_6782_f168v" o="168v" n="336"/>
<pb file="add_6782_f169" o="169" n="337"/>
<div xml:id="echoid-div123" type="page_commentary" level="2" n="123">
<p>
<s xml:id="echoid-s448" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s448" xml:space="preserve">
These three tables show the formulae for the entries in each column of a table generated
from a constant difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, with different patterns ofincreasing and decreasing columns. <lb/>
Increasing columns are indicated by a triangle that broadens at the bottom, thus, Δ;
decreasing columns are indicated by a trinagle that narrows at the bottom. <lb/>
Entries in the first column (after <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>) are denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>1</mn></msup></mrow></mstyle></math>
(where 1 is a superscript, not a power). <lb/>
Entries in the second column are denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>2</mn></msup></mrow></mstyle></math>.
Entries in the third column are denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>3</mn></msup></mrow></mstyle></math>.
Entries in the fourth column are denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>4</mn></msup></mrow></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head64" xml:lang="lat">
Ad progressiones, <lb/>
<lb/>[<emph style="it">tr:
On progressions
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s450" xml:space="preserve">
<sc>
Differences in tables of sines follow the alternately increasing and decreasing pattern given here.
</sc>
ad calculum sinuum
<lb/>[<emph style="it">tr:
for the calculation of sines
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s451" xml:space="preserve">
omnes <reg norm="examinantur" type="abbr">exam</reg>.
<lb/>[<emph style="it">tr:
all examined
</emph>]<lb/>
</s>
</p>
<head xml:id="echoid-head65" xml:lang="lat">
Ad progressiones,
<lb/>[<emph style="it">tr:
On progressioins
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s452" xml:space="preserve">
<sc>
Differences in tables of sines follow the alternately increasing and decreasing pattern given here.
</sc>
ad calculum sinuum
<lb/>[<emph style="it">tr:
for the calculation of sines
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s453" xml:space="preserve">
omnes <reg norm="examinantur" type="abbr">exam</reg>.
<lb/>[<emph style="it">tr:
all examined
</emph>]<lb/>
</s>
</p>
<head xml:id="echoid-head66" xml:lang="lat">
Ad progressiones,
<lb/>[<emph style="it">tr:
On progressions
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s454" xml:space="preserve">
omnes <reg norm="examinantur" type="abbr">exam</reg>. <lb/>
<lb/>[<emph style="it">tr:
all examined
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f169v" o="169v" n="338"/>
<pb file="add_6782_f170" o="170" n="339"/>
<div xml:id="echoid-div124" type="page_commentary" level="2" n="124">
<p>
<s xml:id="echoid-s455" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s455" xml:space="preserve">
At the top right is a numerical table in which every column is decreasing.
The working below the table demonstrates in detail how the entries are calculated. <lb/>
The table below the first double line gives formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries in each column. <lb/>
The table below the second double line gives the same information in a rearranged form.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head67" xml:lang="lat">
Ad progressiones.
<lb/>[<emph style="it">tr:
On progressions
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s457" xml:space="preserve">
omnes <reg norm="examinantur" type="abbr">exam</reg>.
<lb/>[<emph style="it">tr:
all examined
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f170v" o="170v" n="340"/>
<pb file="add_6782_f171" o="171" n="341"/>
<div xml:id="echoid-div125" type="page_commentary" level="2" n="125">
<p>
<s xml:id="echoid-s458" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s458" xml:space="preserve">
At the top right is a numerical example in which the columns alternately increase and decrease.
The working below the table demonstrates in detail how the entries are calculated. <lb/>
The first table below the double line gives formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries in each column. <lb/>
The second table gives the same information but with the fractions rearranged over common denominators.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head68" xml:lang="lat">
Ad progressiones.
<lb/>[<emph style="it">tr:
On progressions
</emph>]<lb/>
</head>
<pb file="add_6782_f171v" o="171v" n="342"/>
<pb file="add_6782_f172" o="172" n="343"/>
<div xml:id="echoid-div126" type="page_commentary" level="2" n="126">
<p>
<s xml:id="echoid-s460" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s460" xml:space="preserve">
At the top of the page is a numerical example in which the columns alternately decrease and increase.
The working demonstrates in detail how the entries are calculated. <lb/>
The small table halfway down on the right contains negative entries in three of its columns. <lb/>
The table at the bottom of the page gives formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries in each column.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head69" xml:lang="lat">
Ad progressiones.
<lb/>[<emph style="it">tr:
On progressions
</emph>]<lb/>
</head>
<pb file="add_6782_f172v" o="172v" n="344"/>
<pb file="add_6782_f173" o="173" n="345"/>
<div xml:id="echoid-div127" type="page_commentary" level="2" n="127">
<p>
<s xml:id="echoid-s462" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s462" xml:space="preserve">
Rules for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries in the third and fourth columns of a table
generated from a constant difference,
for various patterns of increasing and decreasing columns.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f173v" o="173v" n="346"/>
<pb file="add_6782_f174" o="174" n="347"/>
<div xml:id="echoid-div128" type="page_commentary" level="2" n="128">
<p>
<s xml:id="echoid-s464" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s464" xml:space="preserve">
Rules for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries in the third and fourth columns of a table
generated from a constant difference,
for various patterns of increasing and decreasing columns.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f174v" o="174v" n="348"/>
<pb file="add_6782_f175" o="175" n="349"/>
<div xml:id="echoid-div129" type="page_commentary" level="2" n="129">
<p>
<s xml:id="echoid-s466" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s466" xml:space="preserve">
At the top right is a numerical example in which the first three columns increase but the fourth decreases.
The working below the table demonstrates in detail how the entries are calculated. <lb/>
The rule for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entry in the fourth column is given in full, in two different versions. <lb/>
Below the double line, similar rules are given for a table in which the first and second columns increase
but the third and fourth columns decrease.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f175v" o="175v" n="350"/>
<pb file="add_6782_f176" o="176" n="351"/>
<div xml:id="echoid-div130" type="page_commentary" level="2" n="130">
<p>
<s xml:id="echoid-s468" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s468" xml:space="preserve">
General rules for the entries in the third and fourth columns of a table
generated from a constant difference. <lb/>
Instructions for the correct signs are given in a separate note at the end of the page.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head70" xml:lang="lat">
Ad <emph style="st">finales</emph> æquationes <lb/>
generalis methodus
<lb/>[<emph style="it">tr:
On equations, general method
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s470" xml:space="preserve">
In <emph style="st">Descendibus</emph> <emph style="super">Decrescentibus</emph>: adde et subtrahe.
<lb/>[<emph style="it">tr:
In decreasing progressions: add and subtract.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s471" xml:space="preserve">
In Crescentibus: subtrahe et adde.
<lb/>[<emph style="it">tr:
In increasing progressions: subtract and add.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f176v" o="176v" n="352"/>
<pb file="add_6782_f177" o="177" n="353"/>
<div xml:id="echoid-div131" type="page_commentary" level="2" n="131">
<p>
<s xml:id="echoid-s472" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s472" xml:space="preserve">
Rules for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries in the fifth and sixth columns of a table
generated from a constant difference,
for various patterns of increasing and decreasing columns.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f177v" o="177v" n="354"/>
<pb file="add_6782_f178" o="178" n="355"/>
<div xml:id="echoid-div132" type="page_commentary" level="2" n="132">
<p>
<s xml:id="echoid-s474" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s474" xml:space="preserve">
General rules for the entries in the fifth and sixth columns of a table
generated from a constant difference. <lb/>
Instructions for the correct signs are given in a separate note at the end of the page.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s476" xml:space="preserve">
In crescentibus. subtrahe et adde.
<lb/>[<emph style="it">tr:
In increasing porgressions, subtract and add.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s477" xml:space="preserve">
In decrescentibus. adde et subtrahe.
<lb/>[<emph style="it">tr:
In decreasing progressions, add and subtract.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f178v" o="178v" n="356"/>
<div xml:id="echoid-div133" type="page_commentary" level="2" n="133">
<p>
<s xml:id="echoid-s478" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s478" xml:space="preserve">
The leftmost column is marked at the top as increasing (by the triangle widening downwards. Δ),
but after a while it begins to decrease again.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head71" xml:lang="lat">
Nota.
<lb/>[<emph style="it">tr:
Note.
</emph>]<lb/>
</head>
<pb file="add_6782_f179" o="179" n="357"/>
<pb file="add_6782_f179v" o="179v" n="358"/>
<pb file="add_6782_f180" o="180" n="359"/>
<div xml:id="echoid-div134" type="page_commentary" level="2" n="134">
<p>
<s xml:id="echoid-s480" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s480" xml:space="preserve">
This folio shows all combinations of 0, <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>, <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>e</mi></mstyle></math>,
including in each case the null combination consisting only of 0. As on Add MS 6782, f. 35,
each set of combinations is constructed from the previous one
by adding the new letter to the end of each existing combination.
This shows clearly why the number of combinations doubles with each new letter.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f180v" o="180v" n="360"/>
<pb file="add_6782_f181" o="181" n="361"/>
<div xml:id="echoid-div135" type="page_commentary" level="2" n="135">
<p>
<s xml:id="echoid-s482" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s482" xml:space="preserve">
Combinations of two or more quantities, generated by multiplication. <lb/>
The letters <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>, stand for
<foreign xml:lang="lat">pondus</foreign> (weight),
<foreign xml:lang="lat">magnitudo</foreign> (magnitude),
<foreign xml:lang="lat">figura</foreign> (area),
<foreign xml:lang="lat">situs</foreign> (place)
(see Add MS 6786, f. 291).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f181v" o="181v" n="362"/>
<pb file="add_6782_f182" o="182" n="363"/>
<div xml:id="echoid-div136" type="page_commentary" level="2" n="136">
<p>
<s xml:id="echoid-s484" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s484" xml:space="preserve">
This folio lists all the permutations of <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>a</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>,
and begins lists for <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>, <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><mi>f</mi></mstyle></math>. <lb/>
The totals are written as factorials, that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mo>=</mo><mn>1</mn><mo>×</mo><mn>2</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mo>=</mo><mn>1</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>3</mn></mstyle></math>, and so on.
This enables Harriot to calculate the number of permutations of 4 or 5 letters as 120 or 720, respectively,
without writing out the entire list. <lb/>
The calculations at the bottom of the page show how each total is obtained from the previous one;
for example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>4</mn><mo>=</mo><mn>4</mn><mo>×</mo><mn>6</mn></mstyle></math>, and so on.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f182v" o="182v" n="364"/>
<pb file="add_6782_f183" o="183" n="365"/>
<div xml:id="echoid-div137" type="page_commentary" level="2" n="137">
<p>
<s xml:id="echoid-s486" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s486" xml:space="preserve">
The square, cube, and fourth power of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>, possibly in connection with combinations,
which is the predominant subject in the surrounding pages.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f183v" o="183v" n="366"/>
<pb file="add_6782_f184" o="184" n="367"/>
<div xml:id="echoid-div138" type="page_commentary" level="2" n="138">
<p>
<s xml:id="echoid-s488" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s488" xml:space="preserve">
Tables showing all possible throws of 1, 2, or 3 dice.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f184v" o="184v" n="368"/>
<pb file="add_6782_f185" o="185" n="369"/>
<div xml:id="echoid-div139" type="page_commentary" level="2" n="139">
<p>
<s xml:id="echoid-s490" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s490" xml:space="preserve">
This folio is a summary of Harriot's calculations on dice. <lb/>
The tables on the left, after the page has been turned sideways,
are frequency tables for the possible sums that can be obtained by throwing 1, 2, 3, 4, 5, or 6 dice
(see Add MS 6782, f. 41 and f. 40v).
The totals in each case are the appropriate powers of 6. <lb/>
The tables on the right are summaries of the tables that appear on Add MS 6782, f. 50,
and indicate the number of ways repetitions can occur. <lb/>
Amongst the calculations at the bottom right are: <lb/>
(i) a table that appears to convert hours to £, at £30 per hour. <lb/>
(ii) a conversion of 46,656 shillings (one for each possibility for throws of six dice) into £ and shillings. <lb/>
Table (i) is based on £30 or 600 shillings per hour;
converting shillings to throws of the dice, as suggested by table (ii),
gives 600 throws per hour, or 10 throws per minute. <lb/>
(iii) the ratio of throws with repetitions to throws with no repetition, for six dice,
namely (see the table above), <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mn>5</mn><mo>,</mo><mn>9</mn><mn>3</mn><mn>6</mn><mo>:</mo><mn>7</mn><mn>2</mn><mn>0</mn><mo>=</mo><mn>6</mn><mn>3</mn><mfrac><mrow><mn>5</mn><mn>7</mn><mn>6</mn></mrow><mrow><mn>7</mn><mn>2</mn><mn>0</mn></mrow></mfrac><mo>:</mo><mn>1</mn></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f185v" o="185v" n="370"/>
<pb file="add_6782_f186" o="186" n="371"/>
<pb file="add_6782_f186v" o="186v" n="372"/>
<pb file="add_6782_f187" o="187" n="373"/>
<pb file="add_6782_f187v" o="187v" n="374"/>
<pb file="add_6782_f188" o="188" n="375"/>
<pb file="add_6782_f188v" o="188v" n="376"/>
<pb file="add_6782_f189" o="189" n="377"/>
<pb file="add_6782_f189v" o="189v" n="378"/>
<pb file="add_6782_f190" o="190" n="379"/>
<pb file="add_6782_f190v" o="190v" n="380"/>
<pb file="add_6782_f191" o="191" n="381"/>
<pb file="add_6782_f191v" o="191v" n="382"/>
<pb file="add_6782_f192" o="192" n="383"/>
<pb file="add_6782_f192v" o="192v" n="384"/>
<pb file="add_6782_f193" o="193" n="385"/>
<div xml:id="echoid-div140" type="page_commentary" level="2" n="140">
<p>
<s xml:id="echoid-s492" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s492" xml:space="preserve">
A partial draft of page 33 of the 'Magisteria' (Add MS 6782, f. 140).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f193v" o="193v" n="386"/>
<div xml:id="echoid-div141" type="page_commentary" level="2" n="141">
<p>
<s xml:id="echoid-s494" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s494" xml:space="preserve">
Examples from pages 35 and 36 of the 'Magisteria' (Add MS 6782, f. 142 and f. 143).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f194" o="194" n="387"/>
<pb file="add_6782_f194v" o="194v" n="388"/>
<pb file="add_6782_f195" o="195" n="389"/>
<div xml:id="echoid-div142" type="page_commentary" level="2" n="142">
<p>
<s xml:id="echoid-s496" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s496" xml:space="preserve">
A draft of page 31 of the 'Magisteria' (Add MS 6782, f. 138).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head72" xml:space="preserve" xml:lang="lat">
Magisteria.
<lb/>[<emph style="it">tr:
Rules
</emph>]<lb/>
</head>
<pb file="add_6782_f195v" o="195v" n="390"/>
<pb file="add_6782_f196" o="196" n="391"/>
<div xml:id="echoid-div143" type="page_commentary" level="2" n="143">
<p>
<s xml:id="echoid-s498" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s498" xml:space="preserve">
A draft of page 26 of the 'Magisteria' (Add MS 6782, f. 133).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head73" xml:space="preserve" xml:lang="lat">
Pro Magisterio. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
For rules for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s500" xml:space="preserve">
Magisterium. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Rule for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s501" xml:space="preserve">
Hinc apparet quod <emph style="super">hoc</emph>
magisterium fit ex primo numero progressionis et prima <lb/>
differentia, in primo canone, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, scribendo <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><mi>N</mi><mi>N</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi><mi>N</mi><mi>N</mi></mstyle></math>.
<emph style="st">[???]</emph> grad<emph style="super">at</emph>im <lb/>
ut in exemplo.
<lb/>[<emph style="it">tr:
Here it is clear that this rule stems from the first number of the progression and the first difference,
in the first canon, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, writing <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><mi>N</mi><mi>N</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi><mi>N</mi><mi>N</mi></mstyle></math>.
</emph>]<lb/>
</s>
<s xml:id="echoid-s502" xml:space="preserve">
Idem observandum ex <emph style="st">cæteris</emph>
alijs canonibus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, pro <emph style="st">alijs</emph> cæteris <lb/>
magisterijs.
<lb/>[<emph style="it">tr:
The same is to be observed from other canons for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, for the remaining rules.
</emph>]<lb/>
</s>
<s xml:id="echoid-s503" xml:space="preserve">
Similiter agendum pro alijs omnibus Magisterijs <lb/>
cæterarum progressionum.
<emph style="st">mutandis</emph> <emph style="super">notandis</emph>affectionibus secundum speciem.
<lb/>[<emph style="it">tr:
Similarly it may be done for all other rules for the remaining progressions,
noting the sign of the second case.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f196v" o="196v" n="392"/>
<head xml:id="echoid-head74" xml:space="preserve">
Variation of ye needle <lb/>
offered <emph style="super">by</emph> Schouten in his navigation <lb/>
about ye world.
</head>
<p>
<s xml:id="echoid-s504" xml:space="preserve">
To the southeward of the east mouth of the strayts <lb/>
of Magellan in the sight of 57.88 <lb/>
variatio. 12.0. to the NE.
</s>
</p>
<p>
<s xml:id="echoid-s505" xml:space="preserve">
To the southward of the westmost <emph style="super">of Magellane</emph> &amp; 20 legues more <lb/>
westward in 55.43 lat <lb/>
variatio 11.0 (NE).
</s>
</p>
<p>
<s xml:id="echoid-s506" xml:space="preserve">
To the southern altitude 17 &amp; 20 degrees westward <lb/>
from the middest of Magel: straytes, in the common plot <lb/>
of æquall degrees. <lb/>
variatio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> poynt or 6. NW.
</s>
</p>
<p>
<s xml:id="echoid-s507" xml:space="preserve">
20. degrees more westward. lat: [???]: 14.12
variatio. nulla.
</s>
</p>
<pb file="add_6782_f197" o="197" n="393"/>
<div xml:id="echoid-div144" type="page_commentary" level="2" n="144">
<p>
<s xml:id="echoid-s508" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s508" xml:space="preserve">
A partial draft of page 32 of the 'Magisteria' (Add MS 6782, f. 139).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f197v" o="197v" n="394"/>
<pb file="add_6782_f198" o="198" n="395"/>
<div xml:id="echoid-div145" type="page_commentary" level="2" n="145">
<p>
<s xml:id="echoid-s510" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s510" xml:space="preserve">
Rules for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries in four columns of a table generated from a constant difference,
as on Add MS 6782, f 169–176, but here using the notation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> (for sum) instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>v</mi></mstyle></math>,
to indicate that the entries may be thought of as sums (of lower numbered columns)
rather than differences (of higher numbered columns).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head75" xml:lang="lat">
Ad aggregata seu summas progressionum.
<lb/>[<emph style="it">tr:
On the gathering together or sum of progressions.
</emph>]<lb/>
</head>
<pb file="add_6782_f198v" o="198v" n="396"/>
<pb file="add_6782_f199" o="199" n="397"/>
<div xml:id="echoid-div146" type="page_commentary" level="2" n="146">
<p>
<s xml:id="echoid-s512" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s512" xml:space="preserve">
A partial draft of page 32 of the 'Magisteria' (Add MS 6782, f. 139).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s514" xml:space="preserve">
&amp; In Infinitum.
<lb/>[<emph style="it">tr:
etc. Indefinitely.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f199v" o="199v" n="398"/>
<pb file="add_6782_f200" o="200" n="399"/>
<div xml:id="echoid-div147" type="page_commentary" level="2" n="147">
<p>
<s xml:id="echoid-s515" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s515" xml:space="preserve">
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, see page 23 of the 'Magisteria' (Add MS 6782, f. 130).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f200v" o="200v" n="400"/>
<pb file="add_6782_f201" o="201" n="401"/>
<div xml:id="echoid-div148" type="page_commentary" level="2" n="148">
<p>
<s xml:id="echoid-s517" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s517" xml:space="preserve">
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, see page 22 of the 'Magisteria' (Add MS 6782, f. 129).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head76" xml:space="preserve" xml:lang="lat">
Canon <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>
<lb/>[<emph style="it">tr:
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>
</emph>]<lb/>
</head>
<pb file="add_6782_f201v" o="201v" n="402"/>
<pb file="add_6782_f202" o="202" n="403"/>
<div xml:id="echoid-div149" type="page_commentary" level="2" n="149">
<p>
<s xml:id="echoid-s519" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s519" xml:space="preserve">
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, see page 25 of the 'Magisteria' (Add MS 6782, f. 132).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head77" xml:space="preserve" xml:lang="lat">
Canon <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>
</emph>]<lb/>
</head>
<pb file="add_6782_f202v" o="202v" n="404"/>
<pb file="add_6782_f203" o="203" n="405"/>
<div xml:id="echoid-div150" type="page_commentary" level="2" n="150">
<p>
<s xml:id="echoid-s521" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s521" xml:space="preserve">
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, see pages 22 and 25 of the 'Magisteria' (Add MS 6782, f. 129 and f. 132).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head78" xml:space="preserve" xml:lang="lat">
De canone pro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>:
<lb/>[<emph style="it">tr:
On the canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s523" xml:space="preserve">
Videlicet: pro 4<emph style="super">ta</emph> progressione,
3<emph style="super">m</emph> differentiarum gradualium.
<lb/>[<emph style="it">tr:
Clearly set out: for the fourth progression, three grades of differences.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s524" xml:space="preserve">
Cuius <lb/>
species <lb/>
hic
<lb/>[<emph style="it">tr:
For these cases here
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s525" xml:space="preserve">
Et affectio-<lb/>
nes speciem <lb/>
ita.
<lb/>[<emph style="it">tr:
And for these cases of sign thus.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s526" xml:space="preserve">
Sed si species progressiones <lb/>
sit:
<lb/>[<emph style="it">tr:
But if the cases of progressions are:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s527" xml:space="preserve">
Affectiones erunt.
<lb/>[<emph style="it">tr:
The signs will be
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s528" xml:space="preserve">
Etiam, si species <lb/>
sit:
<lb/>[<emph style="it">tr:
Also, if the cases are:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s529" xml:space="preserve">
Affectiones erunt.
<lb/>[<emph style="it">tr:
The signs will be
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s530" xml:space="preserve">
Et similiter de alijs
<lb/>[<emph style="it">tr:
And similarly for others.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f203v" o="203v" n="406"/>
<pb file="add_6782_f204" o="204" n="407"/>
<head xml:id="echoid-head79" xml:space="preserve" xml:lang="lat">
Canon. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>
</emph>]<lb/>
</head>
<pb file="add_6782_f204v" o="204v" n="408"/>
<pb file="add_6782_f205" o="205" n="409"/>
<div xml:id="echoid-div151" type="page_commentary" level="2" n="151">
<p>
<s xml:id="echoid-s531" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s531" xml:space="preserve">
Canons 3 and 4 for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, with columns alternately increasing and decreasing;
see page 24 of the 'Magisteria' (Add MS 6782, f. 131).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head80" xml:space="preserve" xml:lang="lat">
3. Canon <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>
</emph>]<lb/>
</head>
<head xml:id="echoid-head81" xml:space="preserve" xml:lang="lat">
4. Canon <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>
</emph>]<lb/>
</head>
<pb file="add_6782_f205v" o="205v" n="410"/>
<pb file="add_6782_f206" o="206" n="411"/>
<div xml:id="echoid-div152" type="page_commentary" level="2" n="152">
<p>
<s xml:id="echoid-s533" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s533" xml:space="preserve">
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, see page 21 of the 'Magisteria' (Add MS 6782, f. 128).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head82" xml:space="preserve" xml:lang="lat">
De canone ad dividendam progressionum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
On the canon for dividing the progression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s535" xml:space="preserve">
Videlicet; 5<emph style="super">tam</emph> progressionem,
4<emph style="super">or</emph> differentiarum gradualium.
<lb/>[<emph style="it">tr:
Clearly set out: the fifth progression, four grades of differences.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f206v" o="206v" n="412"/>
<pb file="add_6782_f207" o="207" n="413"/>
<div xml:id="echoid-div153" type="page_commentary" level="2" n="153">
<p>
<s xml:id="echoid-s536" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s536" xml:space="preserve">
Canon 1 for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, with all rows increasing;
see page 24 of the 'Magisteria' (Add MS 6782, f. 131).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head83" xml:space="preserve" xml:lang="lat">
1. Canon <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>
</emph>]<lb/>
</head>
<pb file="add_6782_f207v" o="207v" n="414"/>
<pb file="add_6782_f208" o="208" n="415"/>
<div xml:id="echoid-div154" type="page_commentary" level="2" n="154">
<p>
<s xml:id="echoid-s538" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s538" xml:space="preserve">
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, see page 21 of the 'Magisteria' (Add MS 6782, f. 128).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head84" xml:space="preserve" xml:lang="lat">
Canon <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>
</emph>]<lb/>
</head>
<pb file="add_6782_f208v" o="208v" n="416"/>
<pb file="add_6782_f209" o="209" n="417"/>
<div xml:id="echoid-div155" type="page_commentary" level="2" n="155">
<p>
<s xml:id="echoid-s540" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s540" xml:space="preserve">
Canon 2 for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, with all rows decreasing;
see page 24 of the 'Magisteria' (Add MS 6782, f. 131).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head85" xml:space="preserve" xml:lang="lat">
2. Canon <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>
</emph>]<lb/>
</head>
<pb file="add_6782_f209v" o="209v" n="418"/>
<pb file="add_6782_f210" o="210" n="419"/>
<div xml:id="echoid-div156" type="page_commentary" level="2" n="156">
<p>
<s xml:id="echoid-s542" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s542" xml:space="preserve">
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, see pages 19–20 of the 'Magisteria' (Add MS 6782, f. 126 and f. 127).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head86" xml:space="preserve" xml:lang="lat">
1. Canon. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>
</emph>]<lb/>
</head>
<pb file="add_6782_f210v" o="210v" n="420"/>
<div xml:id="echoid-div157" type="page_commentary" level="2" n="157">
<p>
<s xml:id="echoid-s544" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s544" xml:space="preserve">
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, see pages 19–20 of the 'Magisteria' (Add MS 6782, f. 126 and f. 127).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f211" o="211" n="421"/>
<div xml:id="echoid-div158" type="page_commentary" level="2" n="158">
<p>
<s xml:id="echoid-s546" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s546" xml:space="preserve">
Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, see pages 19-20 of the 'Magisteria' (Add MS 6782, f. 126 and f. 127).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head87" xml:space="preserve" xml:lang="lat">
De canone ad dividendam progressionum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
On the canon for dividing the progression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s548" xml:space="preserve">
Videlicet; sextam progressionem,
quinque differentiarum gradualium <lb/>
gradatarum.
<lb/>[<emph style="it">tr:
Clearly set out: the sixth progression, five grades of differences.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f211v" o="211v" n="422"/>
<div xml:id="echoid-div159" type="page_commentary" level="2" n="159">
<p>
<s xml:id="echoid-s549" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s549" xml:space="preserve">
Sign patterns for columns of difference tables. <lb/>
For further examples see page 8 of the 'Magisteria' (Add MS 6782, f. 115).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f212" o="212" n="423"/>
<div xml:id="echoid-div160" type="page_commentary" level="2" n="160">
<p>
<s xml:id="echoid-s551" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s551" xml:space="preserve">
Difference tables showing some possible variations of increasing (c) and decreasing (d) columns. <lb/>
The charts on the right hand side show sign patterns for each column. <lb/>
For further examples see page 8 of the 'Magisteria' (Add MS 6782, f. 115).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f212v" o="212v" n="424"/>
<pb file="add_6782_f213" o="213" n="425"/>
<div xml:id="echoid-div161" type="page_commentary" level="2" n="161">
<p>
<s xml:id="echoid-s553" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s553" xml:space="preserve">
Difference tables showing some possible variations of increasing (c) and decreasing (d) columns. <lb/>
The charts on the right hand side show the sign patterns for each column. <lb/>
For further examples see page 8 of the 'Magisteria' (Add MS 6782, f. 115).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f213v" o="213v" n="426"/>
<pb file="add_6782_f214" o="214" n="427"/>
<div xml:id="echoid-div162" type="page_commentary" level="2" n="162">
<p>
<s xml:id="echoid-s555" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s555" xml:space="preserve">
Difference tables showing some possible variations of increasing (c) and decreasing (d) columns. <lb/>
The charts on the right hand side show the sign patterns for each column. <lb/>
For further examples see page 8 of the 'Magisteria' (Add MS 6782, f. 115).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f214v" o="214v" n="428"/>
<pb file="add_6782_f215" o="215" n="429"/>
<div xml:id="echoid-div163" type="page_commentary" level="2" n="163">
<p>
<s xml:id="echoid-s557" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s557" xml:space="preserve">
Lists of all possible variations of increasing (c) and decreasing (d) columns. <lb/>
See page 8 of the 'Magisteria' (Add MS 6782, f. 115).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f215v" o="215v" n="430"/>
<div xml:id="echoid-div164" type="page_commentary" level="2" n="164">
<p>
<s xml:id="echoid-s559" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s559" xml:space="preserve">
Lists of all possible variations of increasing (c) and decreasing (d) columns. <lb/>
The charts on the right hand side show sign patterns for individual columns. <lb/>
For a full array of such charts see page 8 of the 'Magisteria' (Add MS 6782, f. 115).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f216" o="216" n="431"/>
<div xml:id="echoid-div165" type="page_commentary" level="2" n="165">
<p>
<s xml:id="echoid-s561" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s561" xml:space="preserve">
Difference tables showing some possible variations of increasing (c) and decreasing (d) columns. <lb/>
The charts on the right hand side show the sign patterns for each column. <lb/>
For further examples see page 8 of the 'Magisteria' (Add MS 6782, f. 115).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f216v" o="216v" n="432"/>
<div xml:id="echoid-div166" type="page_commentary" level="2" n="166">
<p>
<s xml:id="echoid-s563" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s563" xml:space="preserve">
Sign patterns for columns of difference tables. <lb/>
For further examples see page 8 of the 'Magisteria' (Add MS 6782, f. 115).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f217" o="217" n="433"/>
<div xml:id="echoid-div167" type="page_commentary" level="2" n="167">
<p>
<s xml:id="echoid-s565" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s565" xml:space="preserve">
See pages 5 to 7 of the 'Magisteria' (Add MS 6782, f. 110 to f. 112), which contain similar numerical tables.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f217v" o="217v" n="434"/>
<pb file="add_6782_f218" o="218" n="435"/>
<div xml:id="echoid-div168" type="page_commentary" level="2" n="168">
<p>
<s xml:id="echoid-s567" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s567" xml:space="preserve">
A draft for page 2 of the 'Magisteria' (Add MS 6782, f. 109). <lb/>
At the end of the page, Harriot suggestes the notation: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mo>,</mo><mi>n</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mo>,</mo><mi>n</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mo>,</mo><mi>n</mi></mstyle></math>, ... inside small boxes,
or better: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>,</mo><mn>7</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>,</mo><mn>6</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>,</mo><mn>5</mn></mstyle></math>, ... inside small boxes,
for what we now write as: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>n</mi><mn>7</mn></msup></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>n</mi><mn>6</mn></msup></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>n</mi><mn>5</mn></msup></mrow></mstyle></math>, ....
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s569" xml:space="preserve">
Magis placet. <lb/>
<lb/>[<emph style="it">tr:
More pleasing.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f218v" o="218v" n="436"/>
<pb file="add_6782_f219" o="219" n="437"/>
<div xml:id="echoid-div169" type="page_commentary" level="2" n="169">
<p>
<s xml:id="echoid-s570" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s570" xml:space="preserve">
The calculations of the previous page (Add MS 6782, f. 218) demonstrated numerically.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f219v" o="219v" n="438"/>
<pb file="add_6782_f220" o="220" n="439"/>
<div xml:id="echoid-div170" type="page_commentary" level="2" n="170">
<p>
<s xml:id="echoid-s572" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s572" xml:space="preserve">
Canons for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, see page 22 of the 'Magisteria' (Add MS 6782, f. 129).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head88" xml:space="preserve" xml:lang="lat">
De Canone. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. De Canone. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
On the canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. On the canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
</emph>]<lb/>
</head>
<pb file="add_6782_f220v" o="220v" n="440"/>
<pb file="add_6782_f221" o="221" n="441"/>
<pb file="add_6782_f221v" o="221v" n="442"/>
<pb file="add_6782_f222" o="222" n="443"/>
<pb file="add_6782_f222v" o="222v" n="444"/>
<pb file="add_6782_f223" o="223" n="445"/>
<pb file="add_6782_f223v" o="223v" n="446"/>
<pb file="add_6782_f224" o="224" n="447"/>
<pb file="add_6782_f224v" o="224v" n="448"/>
<pb file="add_6782_f225" o="225" n="449"/>
<pb file="add_6782_f225v" o="225v" n="450"/>
<pb file="add_6782_f226" o="226" n="451"/>
<pb file="add_6782_f226v" o="226v" n="452"/>
<pb file="add_6782_f227" o="227" n="453"/>
<pb file="add_6782_f227v" o="227v" n="454"/>
<pb file="add_6782_f228" o="228" n="455"/>
<div xml:id="echoid-div171" type="page_commentary" level="2" n="171">
<p>
<s xml:id="echoid-s574" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s574" xml:space="preserve">
A numerical example of the rule for square root of a sum.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f228v" o="228v" n="456"/>
<pb file="add_6782_f229" o="229" n="457"/>
<div xml:id="echoid-div172" type="page_commentary" level="2" n="172">
<p>
<s xml:id="echoid-s576" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s576" xml:space="preserve">
The references on this page are to Viète's
<emph style="it">Supplementum geometriæ</emph> (1593).
</s>
<lb/>
<quote xml:lang="lat">
Propositio III. <lb/>
Si duae lineae rectae Ã  puncto extra circulum eductae ipsum secent,
pars autem exterior primae fit proportionalis inter partem exteriorem secundae &amp; partem interiorem ejusdem:
erit quoque pars exterior secundae proportionalis inter partem exteriorem primae &amp; partem interiorem ejusdem.
</quote>
<lb/>
<quote>
If two straight lines drawn from a point outside a circle cut it in such a way that
the external part of the first is a proportional between the external and internal parts of the second,
the external part of the second will be a proportional between the external and internal parts of the first.
</quote>
<lb/>
<quote xml:lang="lat">
Propositio IV. <lb/>
Si duae lineae rectae Ã  puncto extra circulum eductae ipsum secent
quod autem fit sub partibus exterioribus eductarum, aequale fit ei quod fit sub intertioribus:
exteriores partes permutatim sumptae, erunt continue proportionales inter partes interiors.
</quote>
<lb/>
<quote>
If two straight lines drawn from a point outside a circle cut it,
and moreover the product of the external parts is equal to that of the internal parts,
the external parts taken in turn will be continued proportionals between the internal parts.
</quote>
<lb/>
<quote xml:lang="lat">
Propositio V. <lb/>
Datis duabus lineis rectis, invenire inter easdem duas medias continue, proportionales.
</quote>
<lb/>
<quote>
Given two straight lines, to find two mean proportionals between them.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head89" xml:space="preserve" xml:lang="lat">
Vieta. supl.
pag. 14. b.
<lb/>[<emph style="it">tr:
Viète, Supplementum, page 14v.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s578" xml:space="preserve">
prop. 5.)
<lb/>[<emph style="it">tr:
Proposition 5.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s579" xml:space="preserve">
prop 3.)
<lb/>[<emph style="it">tr:
Proposition 3.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s580" xml:space="preserve">
prop: 4.)
<lb/>[<emph style="it">tr:
Proposition 4.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f229v" o="229v" n="458"/>
<pb file="add_6782_f230" o="230" n="459"/>
<pb file="add_6782_f230v" o="230v" n="460"/>
<pb file="add_6782_f231" o="231" n="461"/>
<div xml:id="echoid-div173" type="page_commentary" level="2" n="173">
<p>
<s xml:id="echoid-s581" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s581" xml:space="preserve">
This sheet refers to Stevin's <emph style="it">L'arithmétique ... aussi l'algebre</emph> (1585), page 215,
where there is a section entitled 'De l'addition des racines de multinomies radicaux'.
Stevin gave numerical examples, whereas Harrot has worked in letters.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s583" xml:space="preserve">
vide: Stevin. 215.
<lb/>[<emph style="it">tr:
See Stevin, page 215.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f231v" o="231v" n="462"/>
<pb file="add_6782_f232" o="232" n="463"/>
<div xml:id="echoid-div174" type="page_commentary" level="2" n="174">
<p>
<s xml:id="echoid-s584" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s584" xml:space="preserve">
Proofs of the inequalities <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>+</mo><mi>c</mi><mi>c</mi><mo>&gt;</mo><mn>2</mn><mi>b</mi><mi>c</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>c</mi><mo>&gt;</mo><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f232v" o="232v" n="464"/>
<pb file="add_6782_f233" o="233" n="465"/>
<pb file="add_6782_f233v" o="233v" n="466"/>
<pb file="add_6782_f234" o="234" n="467"/>
<div xml:id="echoid-div175" type="page_commentary" level="2" n="175">
<p>
<s xml:id="echoid-s586" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s586" xml:space="preserve">
The table at the top gives formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entry in each column of a table generated
from a constant difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, with every column increasing. <lb/>
Entries in column 0 are constant (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>). <lb/>
Entries in column 1 are denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>1</mn></msup></mrow></mstyle></math>. <lb/>
Entries in column 2 are denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>2</mn></msup></mrow></mstyle></math>. <lb/>
and so on; the small numbers are superscripts, not powers. <lb/>
The lower half of the page lists the possible combinations of increreasing columns (c) and decreasing columns (d),
for up to four columns.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head90" xml:lang="lat">
Ad progressiones.
<lb/>[<emph style="it">tr:
On progressions
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s588" xml:space="preserve">
Casus differentiarum progressionum.
<lb/>[<emph style="it">tr:
Cases of progressions of differences.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s589" xml:space="preserve">
c. designat crescentes progressiones:
<lb/>[<emph style="it">tr:
c. denotes increasing progressions
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s590" xml:space="preserve">
d. decrescentes.
<lb/>[<emph style="it">tr:
d. decreasing
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s591" xml:space="preserve">
æquationis secundi omnes <lb/>
hoc casus habentur in <lb/>
alijs chartis.
<lb/>[<emph style="it">tr:
all cases of the second equation here are to be found in other sheets.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f234v" o="234v" n="468"/>
<pb file="add_6782_f235" o="235" n="469"/>
<div xml:id="echoid-div176" type="page_commentary" level="2" n="176">
<p>
<s xml:id="echoid-s592" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s592" xml:space="preserve">
Rules for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries in six columns of a table generated from a constant difference,
as on Add MS 6782, f 177–178, but here using the notation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> (for sum) instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>v</mi></mstyle></math>,
to indicate that the entries may be thought of as sums (of lower numbered columns)
rather than differences (of higher numbered columns).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f235v" o="235v" n="470"/>
<pb file="add_6782_f236" o="236" n="471"/>
<div xml:id="echoid-div177" type="page_commentary" level="2" n="177">
<p>
<s xml:id="echoid-s594" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s594" xml:space="preserve">
Rules for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries in six columns of a table generated from a constant difference,
as on Add MS 6782, f 177–178, but here using the notation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> (for sum) instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>v</mi></mstyle></math>,
to indicate that the entries may be thought of as sums (of lower numbered columns)
rather than differences (of higher numbered columns).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f236v" o="236v" n="472"/>
<pb file="add_6782_f237" o="237" n="473"/>
<div xml:id="echoid-div178" type="page_commentary" level="2" n="178">
<p>
<s xml:id="echoid-s596" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s596" xml:space="preserve">
Calculations similar to those set out on Add MS 6782, f. 148.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head91" xml:space="preserve">
A.
</head>
<p xml:lang="lat">
<s xml:id="echoid-s598" xml:space="preserve">
Ad numeros triangulos et illorum progenies.
<lb/>[<emph style="it">tr:
On triangular numbers and their progeny.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f237v" o="237v" n="474"/>
<pb file="add_6782_f238" o="238" n="475"/>
<pb file="add_6782_f238v" o="238v" n="476"/>
<pb file="add_6782_f239" o="239" n="477"/>
<div xml:id="echoid-div179" type="page_commentary" level="2" n="179">
<p>
<s xml:id="echoid-s599" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s599" xml:space="preserve">
At the top of the page is a numerical table in which the leftmost column contains sums of fourth powers
(1 + 16 + 81 + ...). <lb/>
Below that is a smaller numerical table in which the leftmost column contains sums of squares. <lb/>
Further down on the right is a numerical table in which the leftmost column contains sums of cubes. <lb/>
Each table lists successive differences until a final (constant) difference is reached.
</s>
<lb/>
<s xml:id="echoid-s600" xml:space="preserve">
The reference to Maurolico is to his <emph style="it">Arithmeticorum libri duo</emph> (1575).
Pages 52, 63, and 67 contain tables of several kinds of figurate numbers.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s602" xml:space="preserve">
Ad summam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi><mi>Z</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
For the sum of squares of squares
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s603" xml:space="preserve">
Vide Maurolicum <lb/>
in Arithmeticis <lb/>
pag. 52. <lb/>
63. <lb/>
67.
<lb/>[<emph style="it">tr:
See Maurolico, in his Arithmetic, pages 52, 63, 67.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s604" xml:space="preserve">
Ad summam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
For the sum of cubes.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f239v" o="239v" n="478"/>
<pb file="add_6782_f240" o="240" n="479"/>
<div xml:id="echoid-div180" type="page_commentary" level="2" n="180">
<p>
<s xml:id="echoid-s605" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s605" xml:space="preserve">
The top of the page shows the working out of the formula for the sum of square-squares.
</s>
<lb/>
<s xml:id="echoid-s606" xml:space="preserve">
The proposition quoted from Maurolico is from his
<emph style="it">Arithmeticorum libri duo</emph> (1575),
Proposition 58 (page 25):
</s>
<lb/>
<quote xml:lang="lat">
Omnis trianguli quadratus, aequalis est aggregato cuborum ab unitate
usque ad cubum triangulo collateralem inclusiue sumptorum.
</quote>
<lb/>
<quote>
The square of every triangular number is equal to the sum of cubes from one,
to the cube of the side of the triangular number, all taken together.)
</quote>
<lb/>
<s xml:id="echoid-s607" xml:space="preserve">
Maurolico gives as an example <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>2</mn><mn>5</mn><mo>=</mo><mn>1</mn><mo>+</mo><mn>8</mn><mo>+</mo><mn>2</mn><mn>7</mn><mo>+</mo><mn>6</mn><mn>4</mn><mo>+</mo><mn>1</mn><mn>2</mn><mn>5</mn></mstyle></math>.
</s>
<lb/>
<s xml:id="echoid-s608" xml:space="preserve">
At the bottom of the page are formulae for sums of units, lines, squares, cubes, and square-squares.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head92" xml:lang="lat">
Ad aggregata Z. C. ZZ. &amp;c.
<lb/>[<emph style="it">tr:
Towards the sums of squares, cubes, square-squares, etc.
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s610" xml:space="preserve">
least
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s611" xml:space="preserve">
<reg norm="Maurolico" type="abbr">Maurol</reg>. pag. 25
<lb/>[<emph style="it">tr:
Maurolico page 25
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s612" xml:space="preserve">
omnis triangula quadratus <lb/>
æqualis est aggregato cuborum <lb/>
ab unitate usque ad cubum triangulo <lb/>
collateralem incipio sumptae.
<lb/>[<emph style="it">tr:
the square of every triangluar number is equal to the sum of cubes from one,
to the cube of the corresponding triangular number taken from the beginning.
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s613" xml:space="preserve">
lines æquall.
</s>
<lb/>
<s xml:id="echoid-s614" xml:space="preserve">
lines æqually [???]
</s>
<lb/>
<s xml:id="echoid-s615" xml:space="preserve">
squares æqually [???] <lb/>
in their rootes. &amp;c.
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s616" xml:space="preserve">
Vel: per reductionem.
<lb/>[<emph style="it">tr:
Or: by reduction.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f240v" o="240v" n="480"/>
<pb file="add_6782_f241" o="241" n="481"/>
<pb file="add_6782_f241v" o="241v" n="482"/>
<pb file="add_6782_f242" o="242" n="483"/>
<pb file="add_6782_f242v" o="242v" n="484"/>
<pb file="add_6782_f243" o="243" n="485"/>
<pb file="add_6782_f243v" o="243v" n="486"/>
<div xml:id="echoid-div181" type="page_commentary" level="2" n="181">
<p>
<s xml:id="echoid-s617" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s617" xml:space="preserve">
A generalized table of triangular numbers,
in which each entry is the sum of the entry above it and the entry to the left of it.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f244" o="244" n="487"/>
<pb file="add_6782_f244v" o="244v" n="488"/>
<div xml:id="echoid-div182" type="page_commentary" level="2" n="182">
<p>
<s xml:id="echoid-s619" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s619" xml:space="preserve">
A 28-row difference table with constant third difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>.
The third, second, and first columns 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>o</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>, repsectively.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head93" xml:lang="lat">
Ad Differentias differentiarum. &amp;c.
<lb/>[<emph style="it">tr:
On differences of differences etc.
</emph>]<lb/>
</head>
<pb file="add_6782_f245" o="245" n="489"/>
<pb file="add_6782_f245v" o="245v" n="490"/>
<pb file="add_6782_f246" o="246" n="491"/>
<div xml:id="echoid-div183" type="page_commentary" level="2" n="183">
<p>
<s xml:id="echoid-s621" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s621" xml:space="preserve">
The table at the top of the page shows the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>C</mi><mo>+</mo><mn>1</mn><mi>r</mi></mstyle></math> (in modern notation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>+</mo><mi>x</mi></mstyle></math>)
evaluated for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>.</mo></mstyle></math> To the right are columns of successive differences
as far as the constant difference 6. <lb/>
Below that are two further tables, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>C</mi><mo>+</mo><mn>2</mn><mi>r</mi></mstyle></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>+</mo><mi>x</mi></mstyle></math>) and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>C</mi><mo>+</mo><mn>3</mn><mi>r</mi></mstyle></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>+</mo><mi>x</mi></mstyle></math>).
The last one is extrapolated upwards to include values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo>-</mo><mn>2</mn></mstyle></math>. <lb/>
At the bottom of the page are formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries columns 0 to 4 of a table generated from
a constant difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> in column 0.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f246v" o="246v" n="492"/>
<pb file="add_6782_f247" o="247" n="493"/>
<div xml:id="echoid-div184" type="page_commentary" level="2" n="184">
<p>
<s xml:id="echoid-s623" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s623" xml:space="preserve">
The lower part of this page contains some jottings on binary arithmetic:
addition of 10000 and 10010, multiplication of 101 by 111, and numbers from 1 to 16 in binary
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f247v" o="247v" n="494"/>
<pb file="add_6782_f248" o="248" n="495"/>
<head xml:id="echoid-head94" xml:space="preserve" xml:lang="lat">
A. Data media trium proportionalium et differentia extremorum invenire extremas.
<lb/>[<emph style="it">tr:
Given the mean of three proportionals and the difference of the extremes, find the extremes.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s625" xml:space="preserve">
Zet. 2,3:
<lb/>[<emph style="it">tr:
Zetetic II.3
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s626" xml:space="preserve">
z. 1,1:
<lb/>[<emph style="it">tr:
Zetetic I.1
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s627" xml:space="preserve">
Zet. 2,3: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math> aggregatum extremarum
<lb/>[<emph style="it">tr:
Zetetic II.3: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math> sum of the extrmes
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s628" xml:space="preserve">
z. 1,1: Tum:
<lb/>[<emph style="it">tr:
Zetetic I.1: then:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s629" xml:space="preserve">
Zet. 2,3: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math> aggregatum extremarum
<lb/>[<emph style="it">tr:
Zetetic II.3: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math> sum of the extremes
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s630" xml:space="preserve">
zet. 1,1. tum:
<lb/>[<emph style="it">tr:
Zetetic I.1, then:
</emph>]<lb/>
</s>
</p>
<head xml:id="echoid-head95" xml:space="preserve" xml:lang="lat">
A. Data media trium proportionalium et summa extremorum invenire extremas.
<lb/>[<emph style="it">tr:
Given the mean of three proportionals and the sum of the extremes, find the extremes.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s631" xml:space="preserve">
Tum est <lb/>
supra
<lb/>[<emph style="it">tr:
Then is the above
</emph>]<lb/>
</s>
<s xml:id="echoid-s632" xml:space="preserve">
Ergo: <lb/>
Zet. 2,4.
<lb/>[<emph style="it">tr:
Therefore, Zetetic II.3.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s633" xml:space="preserve">
Zet. 1,1. tum:
<lb/>[<emph style="it">tr:
Zetetic I.1, then:
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f248v" o="248v" n="496"/>
<pb file="add_6782_f249" o="249" n="497"/>
<pb file="add_6782_f249v" o="249v" n="498"/>
<div xml:id="echoid-div185" type="page_commentary" level="2" n="185">
<p>
<s xml:id="echoid-s634" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s634" xml:space="preserve">
On this page Harriot solves <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mo>=</mo><mn>1</mn><mn>0</mn></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mo>=</mo><mn>8</mn></mstyle></math>.
The latter has real roots 4 and 2, but the roots of the former are complex, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mo>±</mo><msqrt><mrow><mo>-</mo><mn>1</mn></mrow></msqrt></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f250" o="250" n="499"/>
<div xml:id="echoid-div186" type="page_commentary" level="2" n="186">
<p>
<s xml:id="echoid-s636" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s636" xml:space="preserve">
At the bottom of this page Harriot solves <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>d</mi><mi>d</mi><mo>=</mo><mn>2</mn><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>,
giving the solutions <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>d</mi><mo>±</mo><msqrt><mrow><mo>-</mo><mi>d</mi><mi>d</mi></mrow></msqrt></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f250v" o="250v" n="500"/>
<p>
<s xml:id="echoid-s638" xml:space="preserve">
Bombell
</s>
</p>
<pb file="add_6782_f251" o="251" n="501"/>
<pb file="add_6782_f251v" o="251v" n="502"/>
<pb file="add_6782_f252" o="252" n="503"/>
<pb file="add_6782_f252v" o="252v" n="504"/>
<pb file="add_6782_f253" o="253" n="505"/>
<pb file="add_6782_f253v" o="253v" n="506"/>
<pb file="add_6782_f254" o="254" n="507"/>
<div xml:id="echoid-div187" type="page_commentary" level="2" n="187">
<p>
<s xml:id="echoid-s639" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s639" xml:space="preserve">
The table at the top of the page shows the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>C</mi><mo>+</mo><mn>1</mn><mi>z</mi><mo>+</mo><mn>1</mn><mi>r</mi></mstyle></math>
(in modern notation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>+</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>+</mo><mi>x</mi></mstyle></math>),
evaluated for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn></mstyle></math>. To the right are columns of successive differences
as far as the constant difference 6. <lb/>
Below that are three further tables for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>C</mi><mo>+</mo><mn>2</mn><mi>z</mi><mo>+</mo><mn>3</mn><mi>r</mi></mstyle></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>+</mo><mn>2</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>3</mn><mi>x</mi></mstyle></math>),
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>C</mi><mo>+</mo><mn>3</mn><mi>z</mi><mo>+</mo><mn>2</mn><mi>r</mi></mstyle></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>+</mo><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>2</mn><mi>x</mi></mstyle></math>), and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>C</mi><mo>+</mo><mn>4</mn><mi>z</mi><mo>+</mo><mn>5</mn><mi>r</mi></mstyle></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>+</mo><mn>4</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>5</mn><mi>x</mi></mstyle></math>).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f254v" o="254v" n="508"/>
<pb file="add_6782_f255" o="255" n="509"/>
<pb file="add_6782_f255v" o="255v" n="510"/>
<pb file="add_6782_f256" o="256" n="511"/>
<div xml:id="echoid-div188" type="page_commentary" level="2" n="188">
<p>
<s xml:id="echoid-s641" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s641" xml:space="preserve">
The reference on this page is to Proposition 16 from Viète's
<emph style="it">Supplementum geometriæ</emph> (1593).
</s>
<lb/>
<quote xml:lang="lat">
Proposition XVI. <lb/>
Si duo triangula fuerint aequicrura singula, &amp; ipsa alterum alteri cruribus aequalia,
angulus autem qui est ad basin secundi sit triplus anguli qui est ad basin primi:
cubus ex base primi, minus triplo solido sub base primi &amp; cruris communis quadrato,
aequalis est solido sub base secundi &amp; ejusdem cruris quadrato.
</quote>
<lb/>
<quote>
If two triangles are each isosceles, the legs of one equal to the legs of the other,
and moreover the angle at the base of the second is three times the angle at the base of the first,
then the cube of the first base, minus three times the product of the base of the first and the square of the common side,
is equal to the product of the second base and the square of the same side.
</quote>
<lb/>
<s xml:id="echoid-s642" xml:space="preserve">
For Harriot's statement of Proposition 16, and a geometric version of the proof, see Add MS 6784, f. 351.
Here he works the proposition algebraically.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head96" xml:space="preserve">
prop. 16. Supplementi.
<lb/>[<emph style="it">tr:
Proposition 16 from the Supplement
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s644" xml:space="preserve">
duplicature <lb/>
cubus <lb/>[<emph style="it">tr:
the cube is doubled.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f256v" o="256v" n="512"/>
<pb file="add_6782_f257" o="257" n="513"/>
<pb file="add_6782_f257v" o="257v" n="514"/>
<pb file="add_6782_f258" o="258" n="515"/>
<pb file="add_6782_f258v" o="258v" n="516"/>
<pb file="add_6782_f259" o="259" n="517"/>
<pb file="add_6782_f259v" o="259v" n="518"/>
<pb file="add_6782_f260" o="260" n="519"/>
<pb file="add_6782_f260v" o="260v" n="520"/>
<pb file="add_6782_f261" o="261" n="521"/>
<pb file="add_6782_f261v" o="261v" n="522"/>
<pb file="add_6782_f262" o="262" n="523"/>
<head xml:id="echoid-head97" xml:space="preserve" xml:lang="lat">
1.) Apotome ex linea secta in extrema et media <lb/>
ratione.
<lb/>[<emph style="it">tr:
An apotome from cutting a line in extreme and mean ratio
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s645" xml:space="preserve">
Apotome 5<emph style="super">ta</emph>.
<lb/>[<emph style="it">tr:
A fifth apotome.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s646" xml:space="preserve">
Apotome 1<emph style="super">a</emph>.
<lb/>[<emph style="it">tr:
A first apotome.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s647" xml:space="preserve">
Apotome 1<emph style="super">a</emph>.
<lb/>[<emph style="it">tr:
A first apotome.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s648" xml:space="preserve">
Apot: 5<emph style="super">a</emph>. Apot: 1<emph style="super">a</emph>.
<lb/>[<emph style="it">tr:
A fifth apotome. A first apotome.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s649" xml:space="preserve">
Bin: 5. Apot: 5.
<lb/>[<emph style="it">tr:
A fifth binome. A fifth apotome.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s650" xml:space="preserve">
Bin: 1<emph style="super">a</emph>. Bin: 5<emph style="super">a</emph>.
<lb/>[<emph style="it">tr:
A first binome. A fifth apotome.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f262v" o="262v" n="524"/>
<pb file="add_6782_f263" o="263" n="525"/>
<div xml:id="echoid-div189" type="page_commentary" level="2" n="189">
<p>
<s xml:id="echoid-s651" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s651" xml:space="preserve">
Continued from Add MS 6782, f. 262.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head98" xml:space="preserve" xml:lang="lat">
2.) De linea secta extrema e media ratione
<lb/>[<emph style="it">tr:
On a line cut in extreme and mean ratio
</emph>]<lb/>
</head>
<pb file="add_6782_f263v" o="263v" n="526"/>
<pb file="add_6782_f264" o="264" n="527"/>
<div xml:id="echoid-div190" type="page_commentary" level="2" n="190">
<p>
<s xml:id="echoid-s653" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s653" xml:space="preserve">
This page contains an analysis of Proposition 1 from Book XIII of Euclid's <emph style="it">Elements</emph>:
</s>
<lb/>
<quote>
XIII.1 If a straight line is cut in extreme and mean ratio,
then the square on the greater segment added to the half of the whole is five times the square on the half.
</quote>
<lb/>
<s xml:id="echoid-s654" xml:space="preserve">
The left hand column gives the 'analysis' or 'resolution' of the problem,
beginning from the final statement and working backwards to discover what conditions must hold. <lb/>
The right hand column gives the 'synthesis' or 'composition',
beginning from the given conditions and working forward to the proof of the proposition.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head99" xml:space="preserve" xml:lang="lat">
Euclid: lib: 13
<lb/>[<emph style="it">tr:
Euclid Book XIII
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s656" xml:space="preserve">
prop. 1. analysis
<lb/>[<emph style="it">tr:
Proposition 1, analysis
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s657" xml:space="preserve">
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> secta extra: &amp; med. <lb/>
in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><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></mstyle></math> be cut in extrme and mean ratio in point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s658" 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-s659" xml:space="preserve">
Resolutio
<lb/>[<emph style="it">tr:
Resolution
</emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s660" xml:space="preserve">
Est igitur <lb/>
est enim:
<lb/>[<emph style="it">tr:
Therefore it is so; for it is so.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s661" xml:space="preserve">
Compositio
<lb/>[<emph style="it">tr:
Composition
</emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s662" xml:space="preserve">
Quod demonstrare <lb/>
oportuit.
<lb/>[<emph style="it">tr:
Which was to be demonstrated.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f264v" o="264v" n="528"/>
<pb file="add_6782_f265" o="265" n="529"/>
<div xml:id="echoid-div191" type="page_commentary" level="2" n="191">
<p>
<s xml:id="echoid-s663" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s663" xml:space="preserve">
The irrationals defined by Euclid in Book X of the <emph style="it">Elements</emph>
are binomes, bimedials, and so on. For their definitions and properties see Add MS 6783, f. 356v to f. 343v.
Here Harriot defines some further irrational quantities, all of them involving fourth roots,
which do not fall into any of Euclid's categories. See also Add MS 6782, f. 266.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head100" xml:space="preserve" xml:lang="lat">
De speciebus irrationalium ab Euclide omissis
<lb/>[<emph style="it">tr:
On types of irrationals missed by Eculid
</emph>]<lb/>
</head>
<pb file="add_6782_f265v" o="265v" n="530"/>
<pb file="add_6782_f266" o="266" n="531"/>
<div xml:id="echoid-div192" type="page_commentary" level="2" n="192">
<p>
<s xml:id="echoid-s665" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s665" xml:space="preserve">
The irrationals defined by Euclid in Book X of the <emph style="it">Elements</emph>
are binomes, bimedials, and so on. For their definitions and properties see Add MS 6783, f. 356v to f. 343v.
Here Harriot defines some further irrational quantities, all of them involving fourth roots,
which do not fall into any of Euclid's categories. See also Add MS 6782, f. 265.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head101" xml:space="preserve" xml:lang="lat">
De speciebus irrationalium ab Euclide omissis
<lb/>[<emph style="it">tr:
On types of irrationals missed by Eculid
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s667" xml:space="preserve">
Nota
<lb/>[<emph style="it">tr:
Note
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s668" xml:space="preserve">
Animadvertendum quod quælibet harum <emph style="st">specier</emph> <emph style="super">irrationalium</emph>
producit quadratum <lb/>
trinomium compositum ex binomio et mediali.
</s>
<s xml:id="echoid-s669" xml:space="preserve">
Et quodlibet bino-<lb/>
mium huius<emph style="super">modi</emph> speciei <emph style="super">logisticæ</emph> continet in se
implicite duas subspecies.
</s>
<s xml:id="echoid-s670" xml:space="preserve">
Quod <lb/>
si <emph style="super">in singulis</emph> distincte explica<emph style="super">tur</emph>,
ex istis 5 irrationalibus fient 10.
</s>
<s xml:id="echoid-s671" xml:space="preserve">
Ut <lb/>
alijs chartis sequentibus apparebit.
<lb/>[<emph style="it">tr:
It is to be noted that any of these irrationals squared produces a trinomial composed of a binome and a medial.
And any binome of this form in letters contains in itself two subforms.
Which if in each case are set out, from these five irrationals there arise 10.
As will appear in the following sheets.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f266v" o="266v" n="532"/>
<pb file="add_6782_f267" o="267" n="533"/>
<div xml:id="echoid-div193" type="page_commentary" level="2" n="193">
<p>
<s xml:id="echoid-s672" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s672" xml:space="preserve">
In modern notation, binomes are numbers of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>m</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>n</mi></mrow></msqrt></mstyle></math>
where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> are integers. <lb/>
In Book X, Definitions II, Euclid defined six kinds of binomes,
according to various relationships of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>,
which for Euclid were geometric lengths (see Heath, III, 5–6 and 101–115).
In modern notation, the six binomes may be defined as follows. <lb/>
Binome 1: a binome of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mo>+</mo><msqrt><mrow><mi>n</mi></mrow></msqrt></mstyle></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mo>&gt;</mo><msqrt><mrow><mi>n</mi></mrow></msqrt></mstyle></math>,
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>m</mi><mn>2</mn></msup></mrow><mo>=</mo><mi>n</mi><mo>+</mo><mi>k</mi></mstyle></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>m</mi></mrow><mrow><msqrt><mrow><mi>k</mi></mrow></msqrt></mrow></mfrac></mstyle></math> is rational; for example <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mo>+</mo><msqrt><mrow><mn>4</mn><mn>8</mn></mrow></msqrt></mstyle></math>. <lb/>
Binome 2: a binome of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>m</mi></mrow></msqrt><mo>+</mo><mi>n</mi></mstyle></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>m</mi></mrow></msqrt><mo>&gt;</mo><mi>n</mi></mstyle></math>,
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mo>=</mo><mrow><msup><mi>n</mi><mn>2</mn></msup></mrow><mo>+</mo><mi>k</mi></mstyle></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><msqrt><mrow><mi>m</mi></mrow></msqrt></mrow><mrow><msqrt><mrow><mi>k</mi></mrow></msqrt></mrow></mfrac></mstyle></math> is rational; for example <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>1</mn><mn>2</mn></mrow></msqrt><mo>+</mo><mn>3</mn></mstyle></math>. <lb/>
Binome 3: a binome of the form<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>m</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>n</mi></mrow></msqrt></mstyle></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>m</mi></mrow></msqrt><mo>&gt;</mo><msqrt><mrow><mi>n</mi></mrow></msqrt></mstyle></math>,
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mo>=</mo><mi>n</mi><mo>+</mo><mi>k</mi></mstyle></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><msqrt><mrow><mi>m</mi></mrow></msqrt></mrow><mrow><msqrt><mrow><mi>k</mi></mrow></msqrt></mrow></mfrac></mstyle></math> is rational; for example <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>8</mn></mrow></msqrt><mo>+</mo><msqrt><mrow><mn>6</mn></mrow></msqrt></mstyle></math>. <lb/>
Binome 4: a binome of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mo>+</mo><msqrt><mrow><mi>n</mi></mrow></msqrt></mstyle></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mo>&gt;</mo><msqrt><mrow><mi>n</mi></mrow></msqrt></mstyle></math>,
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>m</mi><mn>2</mn></msup></mrow><mo>=</mo><mi>n</mi><mo>+</mo><mi>k</mi></mstyle></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>m</mi></mrow><mrow><msqrt><mrow><mi>k</mi></mrow></msqrt></mrow></mfrac></mstyle></math> is non-rational; for example <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mo>+</mo><msqrt><mrow><mn>2</mn></mrow></msqrt></mstyle></math>. <lb/>
Binome 5: a binome of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>m</mi></mrow></msqrt><mo>+</mo><mi>n</mi></mstyle></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>m</mi></mrow></msqrt><mo>&gt;</mo><mi>n</mi></mstyle></math>,
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mo>=</mo><mrow><msup><mi>n</mi><mn>2</mn></msup></mrow><mo>+</mo><mi>k</mi></mstyle></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><msqrt><mrow><mi>m</mi></mrow></msqrt></mrow><mrow><msqrt><mrow><mi>k</mi></mrow></msqrt></mrow></mfrac></mstyle></math> is non-rational; for example <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>+</mo><mn>1</mn></mstyle></math>. <lb/>
Binome 6: a binome of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>m</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>n</mi></mrow></msqrt></mstyle></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>m</mi></mrow></msqrt><mo>&gt;</mo><msqrt><mrow><mi>n</mi></mrow></msqrt></mstyle></math>,
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mo>=</mo><mi>n</mi><mo>+</mo><mi>k</mi></mstyle></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><msqrt><mrow><mi>m</mi></mrow></msqrt></mrow><mrow><msqrt><mrow><mi>k</mi></mrow></msqrt></mrow></mfrac></mstyle></math> is non-rational; for example <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>3</mn></mrow></msqrt><mo>+</mo><msqrt><mrow><mn>2</mn></mrow></msqrt></mstyle></math>. <lb/>
Harriot made two further distinctions for binomes of the fifth and sixth kind according to whether
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math> is itself a square (type i) or not (type (ii). <lb/>
In this and the following folio, Add MS 6782, f. 268,
Harriot shows that the square of any binome is always a binome of the first kind.
This folio shows his working for first, second, and third binomes.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head102" xml:space="preserve" xml:lang="lat">
Binomiorum quadrata, sunt binomia prima.
<lb/>[<emph style="it">tr:
Squares of binomes are binomes of the first kind.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s674" xml:space="preserve">
1. bin)
<lb/>[<emph style="it">tr:
A binomial of the first kind.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s675" xml:space="preserve">
2.)
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s676" xml:space="preserve">
3.)
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s677" xml:space="preserve">
ut supra.
<lb/>[<emph style="it">tr:
as above.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f267v" o="267v" n="534"/>
<pb file="add_6782_f268" o="268" n="535"/>
<div xml:id="echoid-div194" type="page_commentary" level="2" n="194">
<p>
<s xml:id="echoid-s678" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s678" xml:space="preserve">
This folio is the continuation of Add MS 6782, f. 267.
Here Harriot checks that the squares of fourth, fifth, and sixth binomes,
are always binomes of the first kind.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head103" xml:space="preserve" xml:lang="lat">
Binomiorum quadrata, sunt binomia prima.
<lb/>[<emph style="it">tr:
Squares of binomes are binomes of the first kind.
</emph>]<lb/>
</head>
<pb file="add_6782_f268v" o="268v" n="536"/>
<pb file="add_6782_f269" o="269" n="537"/>
<pb file="add_6782_f269v" o="269v" n="538"/>
<pb file="add_6782_f270" o="270" n="539"/>
<pb file="add_6782_f270v" o="270v" n="540"/>
<pb file="add_6782_f271" o="271" n="541"/>
<pb file="add_6782_f271v" o="271v" n="542"/>
<pb file="add_6782_f272" o="272" n="543"/>
<p xml:lang="lat">
<s xml:id="echoid-s680" xml:space="preserve">
Data secundum trium proportionalium: invenire primam et tertiam, <lb/>
ut illarum differentia sit æqualis bis secundæ datæ.
<lb/>[<emph style="it">tr:
Given the second of three proportionals:
find the first and third so that their difference is equal to twice the given second.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s681" xml:space="preserve">
Sit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> prima, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, secunda. <lb/>
ut illarum differentia sit æqualis bis secundæ datæ.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the first, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> the second, such that their difference is equal to twice the given second.
</emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s682" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>a</mi><mi>t</mi><mo>=</mo><mi>a</mi><mi>c</mi></mstyle></math>. Rationalis potentia.
<lb/>[<emph style="it">tr:
Rational in square.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s683" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> Apotome 5<emph style="super">ta</emph>, 1<emph style="super">o</emph>
<lb/>[<emph style="it">tr:
A fifth apotome
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s684" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math> Rationalis posita.
<lb/>[<emph style="it">tr:
The supposed rational
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s685" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>f</mi></mstyle></math> Binomia 5<emph style="super">a</emph>, 1<emph style="super">o</emph>
<lb/>[<emph style="it">tr:
A fifth binome
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s686" xml:space="preserve">
Erit etiam <lb/>[...]<lb/> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math>. Binomia 4<emph style="super">a</emph>.
<lb/>[<emph style="it">tr:
There will also be <lb/>[...]<lb/> a fourth binome
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s687" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>b</mi></mstyle></math>. Apotome 4<emph style="super">a</emph>.
<lb/>[<emph style="it">tr:
a fourth apotome
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s688" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>e</mi></mstyle></math>. cum rationalium medium totum efficiens 1<emph style="super">o</emph>
<lb/>[<emph style="it">tr:
with the rational, making the mean of all
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s689" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi></mstyle></math>. Minor.
<lb/>[<emph style="it">tr:
Lesser
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s690" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>e</mi></mstyle></math>. Media.
<lb/>[<emph style="it">tr:
Mean
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s691" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math>. Rationale et medium potens. 1<emph style="super">o</emph>.
<lb/>[<emph style="it">tr:
A power of the rational and the mean
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f272v" o="272v" n="544"/>
<pb file="add_6782_f273" o="273" n="545"/>
<pb file="add_6782_f273v-274" o="273v-274" n="546"/>
<pb file="add_6782_f273v" o="273v" n="547"/>
<pb file="add_6782_f274" o="274" n="548"/>
<pb file="add_6782_f274v" o="274v" n="549"/>
<pb file="add_6782_f275" o="275" n="550"/>
<pb file="add_6782_f275v" o="275v" n="551"/>
<pb file="add_6782_f276" o="276" n="552"/>
<div xml:id="echoid-div195" type="page_commentary" level="2" n="195">
<p>
<s xml:id="echoid-s692" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s692" xml:space="preserve">
Powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo maxsize="1">)</mo></mstyle></math> up to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mrow><msup><mo maxsize="1">)</mo><mn>5</mn></msup></mrow></mstyle></math>.
Each power is calculated from the previous one by multiplication. <lb/>
Note the use of cossist notation:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> for a first power, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math> for a square, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> for a cube, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>z</mi></mstyle></math> for a square-suare or fourth power,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>ß</mo></mstyle></math> for a sursolid or fifth power. <lb/>
Below the main table is a list of the final sums, including the sixth power (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>c</mi></mstyle></math>),
which has not been calculated on this page
but which can be deduced from the pattern for the previous cases. <lb/>
For a similar table see Add MS 6786, f. 457.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s694" xml:space="preserve">
Forma <emph style="st">generationis continue</emph>
<emph style="super">generandi figurata</emph> <lb/>
<emph style="st">proportionalium ab unitate</emph>
<emph style="super">in binomia radice</emph> <lb/>
per logisticen speciosam: <lb/>
<emph style="st">ad demonstrandum pro parte alium</emph> <lb/>
<emph style="st">in numeris analysin.</emph>
<lb/>[<emph style="it">tr:
A method of generating figurate numbers from binomial roots in letters:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s695" xml:space="preserve">
Nota: pro porismo.
<lb/>[<emph style="it">tr:
Note: for the proof.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s696" xml:space="preserve">
Species partium <lb/>
unius cuisque potentiæ <lb/>
sunt continue proportionales <lb/>
ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
The case of a single part where the powers are in continued proportion as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
</emph>]<lb/>
</s>
<s xml:id="echoid-s697" xml:space="preserve">
Et in numeris, sunt <lb/>
termini minimi si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> sunt primi &amp;c. <lb/>
et non in <lb/>
ratione <lb/>
multiplicant.
<lb/>[<emph style="it">tr:
And in numbers, these are the lowest terms, 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>c</mi></mstyle></math> are the first, <lb/>
and they are not multiplied by some ratio.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f276v" o="276v" n="553"/>
<pb file="add_6782_f277" o="277" n="554"/>
<div xml:id="echoid-div196" type="page_commentary" level="2" n="196">
<p>
<s xml:id="echoid-s698" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s698" xml:space="preserve">
This page shows notation used for first, second, third, ...., ninth powers in the following authors: <lb/>
Diophantus in <emph style="it">Diophanti Alexandrini rerum arithmeticarum libri sex</emph>,
edited by Wilhelm Xylander (1575); <lb/>
François Viète in, for example, <emph style="it">In artem analyticen isagoge</emph> (1591); <lb/>
Bernard Salignac in <emph style="it">Arithmeticae libri duo et algebrae totidem</emph> (1580, 1593); <lb/>
Michael Stifel in <emph style="it">Arithmetica integra</emph> (1544); <lb/>
Christoph Clavius in <emph style="it">Algebra</emph> (1608); <lb/>
Simon Stevin in <emph style="it">L'arithmétique ... aussi l'algèbre</emph> (1585). <lb/>
The inclusion of Clavius in this list is particularly significant since it dates the page to 1608 or later.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s700" xml:space="preserve">
&amp; indices gradarum
<t>
ec. indices of the degrees
</t>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s701" xml:space="preserve">
ut Diophantus et Vieta
<t>
as in Diophantus and Viète
</t>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s702" xml:space="preserve">
ut Salignacus
<t>
as in Salignacus
</t>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s703" xml:space="preserve">
ut Stifelius, Clavius et alij
<t>
as in Stifel, Clavius and others
</t>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s704" xml:space="preserve">
ut Stevinus et alij
<t>
as in Stevin and others
</t>
</s>
</p>
<pb file="add_6782_f277v" o="277v" n="555"/>
<pb file="add_6782_f278" o="278" n="556"/>
<div xml:id="echoid-div197" type="page_commentary" level="2" n="197">
<p>
<s xml:id="echoid-s705" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s705" xml:space="preserve">
Powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mn>6</mn><mn>0</mn><mo>+</mo><mn>7</mn><mo maxsize="1">)</mo></mstyle></math> up to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mn>6</mn><mn>0</mn><mo>+</mo><mn>7</mn><mrow><msup><mo maxsize="1">)</mo><mn>5</mn></msup></mrow></mstyle></math> following the pattern laid out in f. 276. <lb/>
A calculation below each box gives the sum of the figures contained in it.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f278v" o="278v" n="557"/>
<pb file="add_6782_f279" o="279" n="558"/>
<div xml:id="echoid-div198" type="page_commentary" level="2" n="198">
<p>
<s xml:id="echoid-s707" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s707" xml:space="preserve">
The example <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mn>6</mn><mn>0</mn><mo>+</mo><mn>7</mn><mrow><msup><mo maxsize="1">)</mo><mn>3</mn></msup></mrow></mstyle></math> from f. 278 set out to show how the binomial coefficients are used.
Thus, in calculating the cube, for wich the coefficients are 1,3, 3, 1,
the cube of 6 (the relevant digit of 60) is used once,
the square is taken 3 times and multiplied by 7,
while 6 is also taken 3 times and multiplied by the square of 7;
finally the cube of 7 is added once.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s709" xml:space="preserve">
Numerorum dispositio <lb/>
ad figuratorum genesin et <lb/>
analysin demonstrandam.
<lb/>[<emph style="it">tr:
The disposition of the numbers for the generation of figurate numbers and for demonstrating the analysis.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f279v" o="279v" n="559"/>
<pb file="add_6782_f280" o="280" n="560"/>
<div xml:id="echoid-div199" type="page_commentary" level="2" n="199">
<p>
<s xml:id="echoid-s710" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s710" xml:space="preserve">
The upper third of the page contains calculations of powers of 24, up to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mn>2</mn><mn>4</mn><mrow><msup><mo maxsize="1">)</mo><mn>5</mn></msup></mrow></mstyle></math>. <lb/>
The lower third of the page contain calculations of powers of 67 (see also f. 278). <lb/>
In the middle of the page, the binomial coefficients are listed in three different layouts.
The table on the left shows how each row may be calculated by adding two copies of the previous row.
A similar table appears again in the lower right of the page.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f280v" o="280v" n="561"/>
<div xml:id="echoid-div200" type="page_commentary" level="2" n="200">
<p>
<s xml:id="echoid-s712" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s712" xml:space="preserve">
The first units mentioned are bushels, a measure of grain, equivalent to 4 pecks or 8 gallons. <lb/>
The page contains a conversion of 6553600000 bushels per square mile to 10485760 bushels per acre
(1 square mile = 640 acres),
and a conversion of 262144 acres to 409 square miles.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s714" xml:space="preserve">
10485,760 bushelles
</s>
</p>
<p>
<s xml:id="echoid-s715" xml:space="preserve">
262,144 acres
</s>
</p>
<p>
<s xml:id="echoid-s716" xml:space="preserve">
640 acres in <lb/>
a square mile
</s>
</p>
<p>
<s xml:id="echoid-s717" xml:space="preserve">
409 miles square
</s>
</p>
<p>
<s xml:id="echoid-s718" xml:space="preserve">
20 miles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>4</mn></mrow><mrow><mn>5</mn></mrow></mfrac></mstyle></math>
</s>
</p>
<pb file="add_6782_f281" o="281" n="562"/>
<div xml:id="echoid-div201" type="page_commentary" level="2" n="201">
<p>
<s xml:id="echoid-s719" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s719" xml:space="preserve">
Here Harriot calculates the square root of 4489, the cube root of 300763,
the fourth root of 20151121, and the sixth root of 1350125107,
demonstrating that the answer is 67 in each case.
This is the analysis, or taking apart, of what has been constructed on f. 278. <lb/>
Maurolico's treatment of cube roots begins on page 110 of his
<emph style="it">Arithmeticorum libri duo</emph> (1575).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head104" xml:lang="lat">
Analysis:
</head>
<p xml:lang="lat">
<s xml:id="echoid-s721" xml:space="preserve">
Inde: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi><mi>d</mi><mo>=</mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mi>d</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math> <lb/>
ut Maurolicus et nos in alia charta demonstravimus.
<lb/>[<emph style="it">tr:
Whence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi><mi>d</mi><mo>=</mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mi>d</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>, as in Maurolicus and as I have demonstrated in another sheet.
</emph>]<lb/>
<sc>
This note shows an alternative method of calculation, attributed to Maurolico,
in which <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> is replaced by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
An asterisk against the note directs the reader to Maurolico's method of calculation, on the right.
</sc>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s722" xml:space="preserve">
Maurolicus <lb/>
[???]
</s>
</p>
<pb file="add_6782_f281v" o="281v" n="563"/>
<pb file="add_6782_f282" o="282" n="564"/>
<pb file="add_6782_f282v" o="282v" n="565"/>
<pb file="add_6782_f283" o="283" n="566"/>
<pb file="add_6782_f283v" o="283v" n="567"/>
<pb file="add_6782_f284" o="284" n="568"/>
<pb file="add_6782_f284v" o="284v" n="569"/>
<pb file="add_6782_f285" o="285" n="570"/>
<pb file="add_6782_f285v" o="285v" n="571"/>
<p xml:lang="lat">
<s xml:id="echoid-s723" xml:space="preserve">
oppose?? <lb/>
suum coniugatum <lb/>
eosdem habet <lb/>
numeros.
<lb/>[<emph style="it">tr:
their conjugates have the same numbers.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s724" xml:space="preserve">
contrarium est <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mo>-</mo><mo>=</mo><mo>+</mo></mstyle></math>.
<lb/>[<emph style="it">tr:
the opposite is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mo>-</mo><mo>=</mo><mo>+</mo></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s725" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>r</mi></mstyle></math> quando:
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>r</mi></mstyle></math> when:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s726" xml:space="preserve">
ut in <lb/>
2<emph style="super">o</emph> et 4<emph style="super">o</emph> casu <lb/>
et ut in alia <lb/>
charta probatur <lb/>
universaliter.
<lb/>[<emph style="it">tr:
as in the 2nd and 4th cases and as it is proved generally in the other sheet.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s727" xml:space="preserve">
tum <emph style="st">habetur</emph> unum (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>) ex consequenti <lb/>
erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mn>2</mn><mi>r</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
then as a consequence one value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>r</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s728" xml:space="preserve">
alterum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> habetur ex universalis <lb/>
methodo in alia charta.
<lb/>[<emph style="it">tr:
another value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> is to be had by the general method in another sheet.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f286" o="286" n="572"/>
<pb file="add_6782_f286v" o="286v" n="573"/>
<pb file="add_6782_f287" o="287" n="574"/>
<pb file="add_6782_f287v" o="287v" n="575"/>
<pb file="add_6782_f288" o="288" n="576"/>
<pb file="add_6782_f288v" o="288v" n="577"/>
<pb file="add_6782_f289" o="289" n="578"/>
<pb file="add_6782_f289v" o="289v" n="579"/>
<pb file="add_6782_f290" o="290" n="580"/>
<pb file="add_6782_f290v" o="290v" n="581"/>
<pb file="add_6782_f291" o="291" n="582"/>
<pb file="add_6782_f291v" o="291v" n="583"/>
<pb file="add_6782_f292" o="292" n="584"/>
<pb file="add_6782_f292v" o="292v" n="585"/>
<pb file="add_6782_f293" o="293" n="586"/>
<pb file="add_6782_f293v" o="293v" n="587"/>
<pb file="add_6782_f294" o="294" n="588"/>
<pb file="add_6782_f294v" o="294v" n="589"/>
<pb file="add_6782_f295" o="295" n="590"/>
<pb file="add_6782_f295v" o="295v" n="591"/>
<pb file="add_6782_f296" o="296" n="592"/>
<pb file="add_6782_f296v" o="296v" n="593"/>
<pb file="add_6782_f297" o="297" n="594"/>
<pb file="add_6782_f297v" o="297v" n="595"/>
<pb file="add_6782_f298" o="298" n="596"/>
<div xml:id="echoid-div202" type="page_commentary" level="2" n="202">
<p>
<s xml:id="echoid-s729" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s729" xml:space="preserve">
On this folio Harriot gives rules for finding all the parameters of an arithmetic progression
given any three of them. <lb/>
The three parameters supposed given are listed in the second column, headed 'data',
where Harriot runs systematically through all the combinations <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>u</mi><mi>n</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>u</mi><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>u</mi><mi>s</mi></mstyle></math>, and so on. <lb/>
Rules for finding the two remaining quantities in each case are given in the third column, headed 'quaesita'. <lb/>
For further details see Stedall 2007.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head105" xml:lang="lat">
Omnes casus arithmeticæ progressionis simplicis <lb/>
primi ordinis
<lb/>[<emph style="it">tr:
All cases of simple arithmetic progressions of the first order
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s731" xml:space="preserve">
casus. data. quæsita.
<lb/>[<emph style="it">tr:
case. given. sought.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s732" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. primus numeris.
<lb/>[<emph style="it">tr:
first number.
</emph>]<lb/>
</s>
<s xml:id="echoid-s733" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. ultimus.
<lb/>[<emph style="it">tr:
last number.
</emph>]<lb/>
</s>
<s xml:id="echoid-s734" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. differentia. <lb/>
<lb/>[<emph style="it">tr:
difference.
</emph>]<lb/>
</s>
<s xml:id="echoid-s735" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>. numeris locorum.
<lb/>[<emph style="it">tr:
number of places.
</emph>]<lb/>
</s>
<s xml:id="echoid-s736" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>. summa.
<lb/>[<emph style="it">tr:
sum.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s737" xml:space="preserve">
6 et 9 casus qui <lb/>
signantur * <lb/>
soluuntur æquationibus <lb/>
quadraticis
<lb/>[<emph style="it">tr:
Cases 6 and 9 marked thus * are solved by quadratic equations.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s738" xml:space="preserve">
Ex his terminis <lb/>
tribus datis, dantur reliqui.
<lb/>[<emph style="it">tr:
From any three terms given, the rest may be found.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s739" xml:space="preserve">
I. unitas.
<lb/>[<emph style="it">tr:
I. a unit.
</emph>]<lb/>
</s>
<s xml:id="echoid-s740" xml:space="preserve">
II. quadratum unitatis.
<lb/>[<emph style="it">tr:
II. the square of a unit.
</emph>]<lb/>
</s>
<s xml:id="echoid-s741" xml:space="preserve">
III. cubus unitatis.
<lb/>[<emph style="it">tr:
III. the cube of a unit.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f298v" o="298v" n="597"/>
<pb file="add_6782_f299" o="299" n="598"/>
<div xml:id="echoid-div203" type="page_commentary" level="2" n="203">
<p>
<s xml:id="echoid-s742" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s742" xml:space="preserve">
Rough work, and some equations in words for arithmetic progressions.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head106" xml:lang="lat">
De progressione Arithmetica.
<lb/>[<emph style="it">tr:
On arithmetic progressions
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s744" xml:space="preserve">
In Arithmetica progressionis <lb/>
<lb/>[<emph style="it">tr:
In arithmetic progressions
</emph>]<lb/>
</s>
<s xml:id="echoid-s745" xml:space="preserve">
[1.]) Numerus terminorum – 1 = Numerus differentiorum. <lb/>
<lb/>[<emph style="it">tr:
The number of terms – 1 = the number of differences.
</emph>]<lb/>
</s>
<s xml:id="echoid-s746" xml:space="preserve">
[2.]) Maximus terminorum – minimo = Summa differentiorum. <lb/>
<lb/>[<emph style="it">tr:
The greatest term – the least term = the sum of the differences.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s747" xml:space="preserve">
[etc.]
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s748" xml:space="preserve">
Terminus primorum vel ultimorum
<lb/>[<emph style="it">tr:
The term of the first or the last
</emph>]<lb/>
</s>
<s xml:id="echoid-s749" xml:space="preserve">
excessus. <lb/>
<lb/>[<emph style="it">tr:
The excess.
</emph>]<lb/>
</s>
<s xml:id="echoid-s750" xml:space="preserve">
Summa. <lb/>
<lb/>[<emph style="it">tr:
The sum.
</emph>]<lb/>
</s>
<s xml:id="echoid-s751" xml:space="preserve">
Numerus locorum. <lb/>
<lb/>[<emph style="it">tr:
The number of places.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f299v" o="299v" n="599"/>
<pb file="add_6782_f300" o="300" n="600"/>
<p>
<s xml:id="echoid-s752" xml:space="preserve">
The ground player <lb/>
The persepctive player
</s>
</p>
<pb file="add_6782_f300v" o="300v" n="601"/>
<pb file="add_6782_f301" o="301" n="602"/>
<pb file="add_6782_f301v" o="301v" n="603"/>
<pb file="add_6782_f302" o="302" n="604"/>
<pb file="add_6782_f302v" o="302v" n="605"/>
<pb file="add_6782_f303" o="303" n="606"/>
<pb file="add_6782_f303v" o="303v" n="607"/>
<pb file="add_6782_f304" o="304" n="608"/>
<pb file="add_6782_f304v" o="304v" n="609"/>
<pb file="add_6782_f305" o="305" n="610"/>
<pb file="add_6782_f305v" o="305v" n="611"/>
<pb file="add_6782_f306" o="306" n="612"/>
<pb file="add_6782_f306v" o="306v" n="613"/>
<pb file="add_6782_f307" o="307" n="614"/>
<pb file="add_6782_f307v" o="307v" n="615"/>
<pb file="add_6782_f308" o="308" n="616"/>
<pb file="add_6782_f308v" o="308v" n="617"/>
<pb file="add_6782_f309" o="309" n="618"/>
<pb file="add_6782_f309v" o="309v" n="619"/>
<pb file="add_6782_f310" o="310" n="620"/>
<pb file="add_6782_f310v" o="310v" n="621"/>
<pb file="add_6782_f311" o="311" n="622"/>
<div xml:id="echoid-div204" type="page_commentary" level="2" n="204">
<p>
<s xml:id="echoid-s753" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s753" xml:space="preserve">
This page refers to Stevin's <emph style="it">L'arithmétique … aussi l'algebre</emph> (1585), page 331,
where Stevin discusses the equation 1(3) = 6(2) + 400 (in modern notation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>=</mo><mn>6</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>=</mo><mn>4</mn><mn>0</mn><mn>0</mn></mstyle></math>.)
Here Harriot works on the same equation, written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>3</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>4</mn><mn>0</mn><mn>0</mn></mstyle></math>. See also Add MS 6782, f. 311v. <lb/>
The letters S,WL that appear in this page presumably refer to Harriot's friend Sir William Lower.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s755" xml:space="preserve">
a) Stevin. 331
</s>
</p>
<p>
<s xml:id="echoid-s756" xml:space="preserve">
S,WL
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s757" xml:space="preserve">
Ergo species non est universalis.
<lb/>[<emph style="it">tr:
Therefore the rule is not general.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f311v" o="311v" n="623"/>
<div xml:id="echoid-div205" type="page_commentary" level="2" n="205">
<p>
<s xml:id="echoid-s758" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s758" xml:space="preserve">
Further work relating to Add MS 6782, f. 311.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s760" xml:space="preserve">
Stevin. 331
</s>
</p>
<pb file="add_6782_f312" o="312" n="624"/>
<div xml:id="echoid-div206" type="page_commentary" level="2" n="206">
<p>
<s xml:id="echoid-s761" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s761" xml:space="preserve">
Further work based on Stevin's <emph style="it">L 19arithmétique … aussi l 19algebre</emph> (1585), page 331.
Here Harriot works on general equations of the type <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>f</mi><mi>f</mi><mi>f</mi></mstyle></math>. <lb/>
The letters S,WL that appear on this page presumably refer to Harriot's friend Sir William Lower.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s763" xml:space="preserve">
b) Stevin. 331
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s764" xml:space="preserve">
species non universalis S,WL
<lb/>[<emph style="it">tr:
The rule is not universal.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f312v" o="312v" n="625"/>
<pb file="add_6782_f313" o="313" n="626"/>
<div xml:id="echoid-div207" type="page_commentary" level="2" n="207">
<p>
<s xml:id="echoid-s765" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s765" xml:space="preserve">
The reference on this page is to Aulus Gellius, <emph style="it">Noctes atticae</emph>, (first printed 1469).
Chapter 22 of Book II is entitled
'De vento iapyge deque aliorum ventorum vocabulis regionibusque accepta ex Favorini sermonibus'.
There Aulus Gellius names the winds from each direction; Harriot has placed them around the points of a compass.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head107" xml:space="preserve" xml:lang="lat">
Ex Aulo Gellio. lib. 2. cap. 22. pag. 63.
</head>
<pb file="add_6782_f313v" o="313v" n="627"/>
<pb file="add_6782_f314" o="314" n="628"/>
<div xml:id="echoid-div208" type="page_commentary" level="2" n="208">
<p>
<s xml:id="echoid-s767" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s767" xml:space="preserve">
Canonical forms for equation with three or four positive roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head108" xml:space="preserve">
B)
</head>
<pb file="add_6782_f314v" o="314v" n="629"/>
<pb file="add_6782_f315" o="315" n="630"/>
<div xml:id="echoid-div209" type="page_commentary" level="2" n="209">
<p>
<s xml:id="echoid-s769" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s769" xml:space="preserve">
An examination of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>6</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>1</mn><mi>a</mi><mo>=</mo><mn>6</mn></mstyle></math>, which has roots 1, 2, 3.
This is one of several equations with multiple roots treated by Viète in
<emph style="it">De numerosa potestatum resolutione</emph>.
Harriot solves it in full on Add MS 6783, f. 187, and refers to it again in Add MS 6783, f. 188.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head109" xml:space="preserve">
C)
</head>
<pb file="add_6782_f315v" o="315v" n="631"/>
<div xml:id="echoid-div210" type="page_commentary" level="2" n="210">
<p>
<s xml:id="echoid-s771" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s771" xml:space="preserve">
An examination of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>1</mn><mn>2</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>2</mn><mn>9</mn><mi>a</mi><mo>=</mo><mn>1</mn><mn>8</mn></mstyle></math>, which has roots 1, 2, 9.
This is one of several equations with multiple roots treated by Viète in
<emph style="it">De numerosa potestatum resolutione</emph>.
Harriot solves it in full on Add MS 6783, f. 187.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head110" xml:space="preserve">
D)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s773" xml:space="preserve">
Triens coefficientis longituidnis. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math>
<lb/>[<emph style="it">tr:
A third of the longitudinal coefficient
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s774" xml:space="preserve">
Triplum quadratum. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>b</mi><mi>b</mi><mo>+</mo><mn>6</mn><mi>b</mi><mi>d</mi><mo>+</mo><mn>3</mn><mi>d</mi><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Three times the square
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s775" xml:space="preserve">
maius est coefficientibus planis <lb/>
per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>d</mi><mi>d</mi><mo>+</mo><mn>9</mn><mi>d</mi><mi>c</mi><mo>+</mo><mn>9</mn><mi>c</mi><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
greater than the plane coefficient by
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s776" xml:space="preserve">
Duplus cubus e triente. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Twice the cube of a third of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math>
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s777" xml:space="preserve">
maior
<lb/>[<emph style="it">tr:
greater than
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s778" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math> <lb/>
in coefficientibus planibus
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math> times the plane coefficient
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s779" xml:space="preserve">
Excessus maximi laterus supra <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math> fit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>
<lb/>[<emph style="it">tr:
The excess of the greatest side over <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s780" xml:space="preserve">
reliqua duo sunt minora <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
the two remaining are less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math>
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s781" xml:space="preserve">
maxium latus <lb/>
erit
<lb/>[<emph style="it">tr:
the greatest side will be
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s782" xml:space="preserve">
Examinatio. Vide Charta B)
<lb/>[<emph style="it">tr:
Examination. See sheet B.
</emph>]<lb/>
[<emph style="it">Note:
Sheet B is Add MS 6782, f. 314.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s783" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> in <lb/>
coefficientia <lb/>
plana.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> times the plane coefficient
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s784" xml:space="preserve">
Differentia.
<lb/>[<emph style="it">tr:
Difference.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s785" xml:space="preserve">
Nota. Vide K in D.1.
<lb/>[<emph style="it">tr:
Note. See K in D.1.
</emph>]<lb/>
[<emph style="it">Note:
Sheet D.1 is Add MS 6782, f. 316.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6782_f316" o="316" n="632"/>
<head xml:id="echoid-head111" xml:space="preserve">
D.1.
</head>
<p xml:lang="lat">
<s xml:id="echoid-s786" xml:space="preserve">
D Nota K.
<lb/>[<emph style="it">tr:
Note K for sheet D.
</emph>]<lb/>
[<emph style="it">Note:
Sheet D is Add MS 6783, f. 315v.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6782_f316v" o="316v" n="633"/>
<pb file="add_6782_f317" o="317" n="634"/>
<div xml:id="echoid-div211" type="page_commentary" level="2" n="211">
<p>
<s xml:id="echoid-s787" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s787" xml:space="preserve">
An examination of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>9</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>2</mn><mn>4</mn><mi>a</mi><mo>=</mo><mn>2</mn><mn>0</mn></mstyle></math>, which has roots 2, 2, 5.
This is one of several equations with multiple roots treated by Viète in
<emph style="it">De numerosa potestatum resolutione</emph>.
Harriot solves it in full on Add MS 6783, f. 187.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head112" xml:space="preserve">
E)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s789" xml:space="preserve">
Triens coeff: long: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>
<lb/>[<emph style="it">tr:
A third of the longitudinal coefficient
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s790" xml:space="preserve">
Triplum quadrat: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>b</mi><mi>b</mi><mo>+</mo><mn>6</mn><mi>b</mi><mi>c</mi><mo>+</mo><mn>3</mn><mi>c</mi><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Three times the square
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s791" xml:space="preserve">
maius est coefficientibus planis <lb/>
per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>c</mi><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
greater than the plane coefficient by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>c</mi><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s792" xml:space="preserve">
Duplus cubus e triente. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Twice the cube of the third
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s793" xml:space="preserve">
maior
</s>
<lb/>[<emph style="it">tr:
greater than
</emph>]<lb/>
<lb/>
<s xml:id="echoid-s794" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> in <lb/>
coefficientibus planibus
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> times the plane coefficient
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s795" xml:space="preserve">
Excessus maximi laterus supra <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>
<lb/>[<emph style="it">tr:
Let the excess of the greatest side over <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s796" xml:space="preserve">
reliqua duo sunt minora <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
the remaining two are less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math>
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s797" xml:space="preserve">
maxium latus <lb/>
erit
<lb/>[<emph style="it">tr:
the greatest side will be
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s798" xml:space="preserve">
Examinatio.
<lb/>[<emph style="it">tr:
Examination
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s799" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> in <lb/>
coeff
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> times the coefficient
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s800" xml:space="preserve">
Differentia.
<lb/>[<emph style="it">tr:
Difference
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s801" xml:space="preserve">
Aliter casus
<lb/>[<emph style="it">tr:
Another case
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f317v" o="317v" n="635"/>
<pb file="add_6782_f318" o="318" n="636"/>
<div xml:id="echoid-div212" type="page_commentary" level="2" n="212">
<p>
<s xml:id="echoid-s802" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s802" xml:space="preserve">
An examination of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>1</mn><mn>8</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>9</mn><mn>5</mn><mi>a</mi><mo>=</mo><mn>1</mn><mn>2</mn><mn>6</mn></mstyle></math>, which has roots 2, 7, 9.
This is one of several equations with multiple roots treated by Viète in
<emph style="it">De potestatum numerosa resolutione</emph>.
Harriot solves it in full on Add MS 6783, f. 187.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head113" xml:space="preserve">
F)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s804" xml:space="preserve">
Triens coefficientis longituidnis. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>d</mi></mstyle></math>
<lb/>[<emph style="it">tr:
A third of the longitudinal coefficient
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s805" xml:space="preserve">
Triplum quadratum. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>b</mi><mi>b</mi><mo>-</mo><mn>6</mn><mi>b</mi><mi>d</mi><mo>+</mo><mn>3</mn><mi>d</mi><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Three times the square
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s806" xml:space="preserve">
maius est coefficientibus planis <lb/>
per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>d</mi><mi>d</mi><mo>+</mo><mn>9</mn><mi>c</mi><mi>c</mi><mo>-</mo><mn>9</mn><mi>c</mi><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
greater than the plane coefficient by
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s807" xml:space="preserve">
Duplus cubus e triente. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Twice the cube of the third
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s808" xml:space="preserve">
minor
<lb/>[<emph style="it">tr:
less than
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s809" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>d</mi></mstyle></math> <lb/>
in coefficientibus planibus
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>d</mi></mstyle></math> times the plane coefficient
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s810" xml:space="preserve">
Excessus
<lb/>[<emph style="it">tr:
Excess
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s811" xml:space="preserve">
medium et maximum latus <lb/>
excedunt. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
medium and maximum sides exceed
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s812" xml:space="preserve">
sit unus vel alter excessu, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
let one or other excess be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s813" xml:space="preserve">
erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mi>d</mi></mstyle></math>, excessus medij 1. adde <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>d</mi></mstyle></math> erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> medium. <lb/>
erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mn>3</mn><mi>c</mi><mo>-</mo><mn>2</mn><mi>d</mi></mstyle></math>, excessus maximi 3. adde <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>d</mi></mstyle></math> erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mn>3</mn><mi>c</mi><mo>-</mo><mn>2</mn><mi>d</mi></mstyle></math> maximum.
<lb/>[<emph style="it">tr:
if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mi>d</mi></mstyle></math>, the excess of the medium, add <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>d</mi></mstyle></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> will be the medium; <lb/>
if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mn>3</mn><mi>c</mi><mo>-</mo><mn>2</mn><mi>d</mi></mstyle></math>, the excess of the maximum, add <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>d</mi></mstyle></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mn>3</mn><mi>c</mi><mo>-</mo><mn>3</mn><mi>d</mi></mstyle></math> will be the maximum;
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s814" xml:space="preserve">
Examinatio
<lb/>[<emph style="it">tr:
Examination
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s815" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>c</mi></mstyle></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>d</mi></mstyle></math> <lb/>
in coeff planis
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>c</mi></mstyle></math> times <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>d</mi></mstyle></math> times the plane coefficient
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s816" xml:space="preserve">
cubus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>c</mi><mo>-</mo><mn>2</mn><mi>d</mi></mstyle></math>
<lb/>[<emph style="it">tr:
The cube of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>c</mi><mo>-</mo><mn>2</mn><mi>d</mi></mstyle></math>
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s817" xml:space="preserve">
Differentia
<lb/>[<emph style="it">tr:
Difference
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s818" xml:space="preserve">
Operationes sunt <lb/>
in dorso D.1.
<lb/>[<emph style="it">tr:
The working is on the back of D.1.
</emph>]<lb/>
[<emph style="it">Note:
The back of sheet D.1 is Add MS 6782, f. 316v.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6782_f318v" o="318v" n="637"/>
<pb file="add_6782_f319" o="319" n="638"/>
<div xml:id="echoid-div213" type="page_commentary" level="2" n="213">
<p>
<s xml:id="echoid-s819" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s819" xml:space="preserve">
An examination of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>6</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>2</mn><mi>a</mi><mo>=</mo><mn>8</mn></mstyle></math>, which has roots 2, 2, 2.
This is one of several equations with multiple roots treated by Viète in
<emph style="it">De potestatum numerosa resolutione</emph>.
Harriot solves it in full on Add MS 6783, f. 187, and refers to it again in Add MS 6783, f. 188.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head114" xml:space="preserve">
G)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s821" xml:space="preserve">
Triens coeff: long: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>
<lb/>[<emph style="it">tr:
A third of the longitudinal coefficient
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s822" xml:space="preserve">
Triplum quadrat: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>b</mi><mi>b</mi><mo>=</mo><mn>3</mn><mi>b</mi><mi>b</mi></mstyle></math> coeff. planis.
<lb/>[<emph style="it">tr:
Three times the square is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>b</mi><mi>b</mi><mn>4</mn><mo>,</mo><mi>t</mi><mi>h</mi><mi>e</mi><mi>p</mi><mi>l</mi><mi>a</mi><mi>n</mi><mi>e</mi><mi>c</mi><mi>o</mi><mi>e</mi><mi>f</mi><mi>f</mi><mi>i</mi><mi>c</mi><mi>i</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo>.</mo></mstyle></math></emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s823" xml:space="preserve">
Duplus cubus e triente. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>b</mi><mi>b</mi><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Twice the cube of the third
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s824" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> <lb/>
in coeff: planib.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> times the plane coefficient
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s825" xml:space="preserve">
Tria latera igitur
<lb/>[<emph style="it">tr:
Therefore the three sides are
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f319v" o="319v" n="639"/>
<pb file="add_6782_f320" o="320" n="640"/>
<div xml:id="echoid-div214" type="page_commentary" level="2" n="214">
<p>
<s xml:id="echoid-s826" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s826" xml:space="preserve">
This set of pages, lettered <emph style="it">aa</emph> to <emph style="it">au</emph> is connected to
Harriot's treatise 'De generatione aequationum canonicarum' in Add MS 6783, f. 183 to f. 163. <lb/>
On this first page, Harriot works out the multiplications
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>c</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>c</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>c</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>a</mi><mi>a</mi><mo>-</mo><mi>b</mi><mi>b</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>a</mi><mi>a</mi><mo>-</mo><mi>b</mi><mi>b</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>a</mi><mi>a</mi><mo>+</mo><mi>b</mi><mi>b</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>.
In each case he writes down the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> that reduce the resulting product to zero.
For the second multiplication, for instance, he shows that the product becomes zero when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math>
but not when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>d</mi></mstyle></math>. <lb/>
For similar content see Add MS 6782, f. 182 (d.2), f. 181 (d.3), f. 180 (d.4), and f. 178 (d.6).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head115" xml:space="preserve">
aa)
</head>
<pb file="add_6782_f320v" o="320v" n="641"/>
<pb file="add_6782_f321" o="321" n="642"/>
<div xml:id="echoid-div215" type="page_commentary" level="2" n="215">
<p>
<s xml:id="echoid-s828" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s828" xml:space="preserve">
This page contains multiplications similar to those on the previous page (Add MS 6782, f. 320)
but now with the terms written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>-</mo><mi>a</mi><mo maxsize="1">)</mo></mstyle></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo maxsize="1">)</mo></mstyle></math>, and so on.
Both versions are treated in Add MS 6782, f. 182 (d.2).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head116" xml:space="preserve">
ab)
</head>
<pb file="add_6782_f321v" o="321v" n="643"/>
<pb file="add_6782_f322" o="322" n="644"/>
<head xml:id="echoid-head117" xml:space="preserve">
ac)
</head>
<div xml:id="echoid-div216" type="page_commentary" level="2" n="216">
<p>
<s xml:id="echoid-s830" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s830" xml:space="preserve">
A treatment of the equation arising from the multiplication <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>c</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>, with a numerical example.
For a more detailed treatment of the same equation see Add MS 6782, f. 181 (d.3).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s832" xml:space="preserve">
Fundamentum
<lb/>[<emph style="it">tr:
Foundation
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f322v" o="322v" n="645"/>
<pb file="add_6782_f323" o="323" n="646"/>
<div xml:id="echoid-div217" type="page_commentary" level="2" n="217">
<p>
<s xml:id="echoid-s833" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s833" xml:space="preserve">
A continuation from Add MS 6782, f. 322 of work on the equation arising from the multiplication
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>c</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>. <lb/>
Harriot states without proof the special form the equation will take when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>=</mo><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>,
when the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> vanishes. For a full derivation see Add MS 6782, f. 181 (d.3).
Harriot calls this form of the cubic equation an 'elliptic' or 'Bombellian' equation.
The special case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mi>c</mi></mstyle></math> he calls 'parabolic'.
For Harriot's definitions of the hyperbolic, elliptic, and parabolic forms
of a cubic equation without a square term, see Add MS 6783, f. 106 (e.8). <lb/>
On this page Harriot also gives the form the equation will take when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>d</mi><mo>+</mo><mi>c</mi><mi>d</mi></mstyle></math>,
when the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> vanishes. For a full derivation see Add MS 6782, f. 181 (d.3).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head118" xml:space="preserve">
ad)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s835" xml:space="preserve">
In charta ac)
<lb/>[<emph style="it">tr:
In sheet <emph style="it">ac</emph>
</emph>]<lb/>
[<emph style="it">Note:
Sheet ac is Add MS 6782, f. 322.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s836" xml:space="preserve">
Eliptica. <lb/>
seu Bombellica si <lb/>
convertitur.
<lb/>[<emph style="it">tr:
Elliptic, or the Bombellian kind if the signs are changed.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s837" xml:space="preserve">
Vide D.)
<lb/>[<emph style="it">tr:
See D.)
</emph>]<lb/>
[<emph style="it">Note:
Sheet D. is Add MS 6783, f. 272.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s838" xml:space="preserve">
æquatio parabolica.
<lb/>[<emph style="it">tr:
parabolic equation
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s839" xml:space="preserve">
parabolica
<lb/>[<emph style="it">tr:
parabolic
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s840" xml:space="preserve">
solummodo
<lb/>[<emph style="it">tr:
only
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f323v" o="323v" n="647"/>
<pb file="add_6782_f324" o="324" n="648"/>
<div xml:id="echoid-div218" type="page_commentary" level="2" n="218">
<p>
<s xml:id="echoid-s841" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s841" xml:space="preserve">
On this page Harriot works with the multiplication from Add MS 6782, f. 322 and f. 323, namely,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>c</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>, but now with the signs changed so that it becomes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>c</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>.
As in Add MS 6782, f. 323, he gives the special form of the equation that arises when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>=</mo><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>,
when the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> vanishes. For a full derivation see Add MS 6782, f. 180 (d.4).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head119" xml:space="preserve" xml:lang="lat">
ae) Conversiones
<lb/>[<emph style="it">tr:
Changes of sign
</emph>]<lb/>
</head>
<pb file="add_6782_f324v" o="324v" n="649"/>
<pb file="add_6782_f325" o="325" n="650"/>
<pb file="add_6782_f325v" o="325v" n="651"/>
<head xml:id="echoid-head120" xml:space="preserve">
f.8
</head>
<pb file="add_6782_f326" o="326" n="652"/>
<pb file="add_6782_f326v" o="326v" n="653"/>
<head xml:id="echoid-head121" xml:space="preserve">
f.8
</head>
<pb file="add_6782_f327" o="327" n="654"/>
<pb file="add_6782_f327v" o="327v" n="655"/>
<head xml:id="echoid-head122" xml:space="preserve">
f.8
</head>
<pb file="add_6782_f328" o="328" n="656"/>
<pb file="add_6782_f328v" o="328v" n="657"/>
<pb file="add_6782_f329" o="329" n="658"/>
<pb file="add_6782_f329v" o="329v" n="659"/>
<p xml:lang="lat">
<s xml:id="echoid-s843" xml:space="preserve">
Archimedes de quadrat: parabola <lb/>
prop: 23. pa: 21.
<lb/>[<emph style="it">tr:
Archimedes, De quadratura parabola, Proposition 23, page 21.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s844" xml:space="preserve">
decrescentes
<lb/>[<emph style="it">tr:
decreasing
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s845" xml:space="preserve">
crescentes
<lb/>[<emph style="it">tr:
increasing
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f330" o="330" n="660"/>
<div xml:id="echoid-div219" type="page_commentary" level="2" n="219">
<p>
<s xml:id="echoid-s846" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s846" xml:space="preserve">
This page is a continuation of Add MS 6783, f. 44v. <lb/>
Note 3 gives the triangular numbers in general algebraic notation:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></mfrac></mstyle></math>, <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>1</mn><mo>×</mo><mn>2</mn></mrow></mfrac></mstyle></math>, <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><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo maxsize="1">)</mo></mrow><mrow><mn>1</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>3</mn></mrow></mfrac></mstyle></math>,
<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><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>3</mn><mo maxsize="1">)</mo></mrow><mrow><mn>1</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>4</mn></mrow></mfrac></mstyle></math>. <lb/>
On the right these formula are given labels that in modern subscript notation would be
<emph style="it">p</emph><emph style="sub">1</emph>, <emph style="it">p</emph><emph style="sub">2</emph>,
<emph style="it">p</emph><emph style="sub">3</emph>, and so on. <lb/>
In the fourth notation, in the lower half of the page,
<emph style="it">p</emph> has been replaced by <emph style="it">v</emph>,
and the terms have been multiplied out to give a one-line expression (or in Harriot's terms, an equation)
instead of a fraction. <lb/>
See also page 1 of the 'Magisteria' (Add MS 6782, f. 108).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s848" xml:space="preserve">
3. Generalis notatio <lb/>
triangularium <lb/>
in notis generalibus.
<lb/>[<emph style="it">tr:
3. General notation for triangular numbers in general symbols.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s849" xml:space="preserve">
Melius ad continuam <lb/>
additionem triangularium.
<lb/>[<emph style="it">tr:
Better for continual addition of triangular numbers.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s850" xml:space="preserve">
4. Quarta notatio <lb/>
per æquationes.
<lb/>[<emph style="it">tr:
4. Fourth notation, by means of equations.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f330v" o="330v" n="661"/>
<div xml:id="echoid-div220" type="page_commentary" level="2" n="220">
<p>
<s xml:id="echoid-s851" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s851" xml:space="preserve">
This page contains two magic squares. <lb/>
It also shows the digits 0 to 9 written in their Arabic form and in characters composed only of straight lines
(see also Add MS 6782, f. 30v).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f331" o="331" n="662"/>
<div xml:id="echoid-div221" type="page_commentary" level="2" n="221">
<p>
<s xml:id="echoid-s853" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s853" xml:space="preserve">
This appears to be the 'other paper' referred to on Add MS 6782, f. 38,
since the table at the top of this page is the same as the one that appears there. <lb/>
On the right, the first few entries from the third and fourth columns are written in factorial form,
showing why the ratios of the entries in the first two rows are 1 : 6 and 2 : 5.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head123" xml:space="preserve">
Of combinations.
</head>
<p xml:lang="lat">
<s xml:id="echoid-s855" xml:space="preserve">
Questi minuti numerator <lb/>
multiplicatur per 2. <lb/>
&amp; denominator per 5.
<lb/>[<emph style="it">tr:
The numerator of these fractions is multiplied by 2 etc., the denominator by 5.
</emph>]<lb/>
</s>
<s xml:id="echoid-s856" xml:space="preserve">
Ergo tertius et quartus <lb/>
habent ratione ut 2 ad 5.
<lb/>[<emph style="it">tr:
Therefore the third and the fourth have a ratio of 2 to 5.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s857" xml:space="preserve">
unde ratio in omnibus.
<lb/>[<emph style="it">tr:
whence the ratio in all of them.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f331v" o="331v" n="663"/>
<pb file="add_6782_f332" o="332" n="664"/>
<div xml:id="echoid-div222" type="page_commentary" level="2" n="222">
<p>
<s xml:id="echoid-s858" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s858" xml:space="preserve">
A table of factorials from 1! = 1, to 25! = 15,511,210,043,330,985,984,000,000.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head124" xml:space="preserve">
For Transpositions.
</head>
<pb file="add_6782_f332v" o="332v" n="665"/>
<pb file="add_6782_f333" o="333" n="666"/>
<pb file="add_6782_f333v" o="333v" n="667"/>
<pb file="add_6782_f334" o="334" n="668"/>
<pb file="add_6782_f334v" o="334v" n="669"/>
<pb file="add_6782_f335" o="335" n="670"/>
<div xml:id="echoid-div223" type="page_commentary" level="2" n="223">
<p>
<s xml:id="echoid-s860" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s860" xml:space="preserve">
The same information as in Add MS 6782, f. 336, now presented slightly differently.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head125" xml:lang="lat">
Progressiones crescentes; quarum principia sunt quivis numeri.
<lb/>[<emph style="it">tr:
Increasing progressions; of which the first terms are any numbers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s862" xml:space="preserve">
Melior forma <lb/>
sive optime.
<lb/>[<emph style="it">tr:
A better form, perhaps the best.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f335v" o="335v" n="671"/>
<pb file="add_6782_f336" o="336" n="672"/>
<div xml:id="echoid-div224" type="page_commentary" level="2" n="224">
<p>
<s xml:id="echoid-s863" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s863" xml:space="preserve">
General formulae for the entries in a table generated from a constant difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>,
where every column is increasing (as signified by the symbols Δ above each column). <lb/>
The first entry in column 1 is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. <lb/>
The first entry in column 2 is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>2</mn></msup></mrow></mstyle></math> (where 2 is a superscript, not a power). <lb/>
The first entry in column 3 is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>3</mn></msup></mrow></mstyle></math> (where 3 is a superscript, not a power). <lb/>
And so on.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head126" xml:space="preserve" xml:lang="lat">
progressiones crescentes, quarum principia <lb/>
sunt quibus numeri.
<lb/>[<emph style="it">tr:
increasing progressions, of which the first terms are any numbers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s865" xml:space="preserve">
Vide meliorem <lb/>
formam in alia <lb/>
Charta.
<lb/>[<emph style="it">tr:
See a better form in the other sheet.
</emph>]<lb/>
<sc>
The other sheet mentioned here is Add MS 6782, f. 355.
</sc>
</s>
</p>
<pb file="add_6782_f336v" o="336v" n="673"/>
<pb file="add_6782_f337" o="337" n="674"/>
<div xml:id="echoid-div225" type="page_commentary" level="2" n="225">
<p>
<s xml:id="echoid-s866" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s866" xml:space="preserve">
These are the symbols Harriot devised in 1585 for writing down the native Indian language of Algonquin.
In the right-hand column there are 12 vowels followed by 24 consonants.
In the left-hand column are words representing each sound.
The sounds and then the words are transcribed here from Harriot's notes on another copy of this page,
now held at Westminster School, London, and reproduced in Stedall 2007.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s868" xml:space="preserve">
[sounds] <lb/>
as (a) in all, tall, fall, call <lb/>
as (o) in ore, for, core <lb/>
as (a) in arrow, man, pan <lb/>
as (u) in us, upon, but, cut <lb/>
as (a) in ape, ale, any , are <lb/>
as (e) in erbe, end, the <lb/>
as (i) in ise, ire, pipe <lb/>
as (e) in he, shee, or (ee) in thee, eele. <lb/>
[penultimate pair of vowels]
(in barbarouse wordes only and not to be expressed viva voce.) <lb/>
as (o) in so, no, otes. <lb/>
as (o) in do, to, shoe <lb/>
as (y) in yea, yes, day <lb/>
as (w) in way, was, now, sow <lb/>
as (r) in roote <lb/>
as (l) in lake <lb/>
as (z) in zone, zachary <lb/>
as the French (i) in je, jeter, or as (g) in hodge, iudge <lb/>
as (s) in sault, samon <lb/>
as (sh) in she, shoe <lb/>
as (m) in man <lb/>
as (n) in not <lb/>
as (ng) in king, fling, thing (or (n) in knave) <lb/>
as (v) in vine, geve <lb/>
as (th) in the, thine, there <lb/>
as (gh) in some barabrouse wordes <lb/>
as (f) in fling, fear, of <lb/>
as (th) in thing, thorne <lb/>
as (ch) in some barbarouse wordes or as the Greeke <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>χ</mi></mstyle></math> <lb/>
as (h) in hat, he, oh <lb/>
as (b) in borde, but <lb/>
as (d) in do, grudge, good <lb/>
as (g) in good, god, geve, hog <lb/>
as (p) in pan <lb/>
as (t) in tooth, to, ten, hat <lb/>
as (c) in corne or as (k) in keepe
</s>
</p>
<p>
<s xml:id="echoid-s869" xml:space="preserve">
[words] <lb/>
armes. ore. <lb/>
arow. urchin. <lb/>
aye. err. <lb/>
ice. eele. <lb/>
(in barbarous words only and not to be expressed viva voce.) <lb/>
oates. oon. <lb/>
ye. ne. <lb/>
root. lake. <lb/>
zone. je. <lb/>
sault. shoo. <lb/>
men. nete. gna. <lb/>
vine. thing. ghi. <lb/>
flinge. thorne. chi. <lb/>
bore. drudge. gold. <lb/>
pan. toothe corne.
</s>
</p>
<pb file="add_6782_f337v" o="337v" n="675"/>
<pb file="add_6782_f338" o="338" n="676"/>
<pb file="add_6782_f338v" o="338v" n="677"/>
<pb file="add_6782_f339" o="339" n="678"/>
<pb file="add_6782_f339v" o="339v" n="679"/>
<pb file="add_6782_f340" o="340" n="680"/>
<pb file="add_6782_f340v" o="340v" n="681"/>
<pb file="add_6782_f341" o="341" n="682"/>
<pb file="add_6782_f341v" o="341v" n="683"/>
<pb file="add_6782_f342" o="342" n="684"/>
<pb file="add_6782_f342v" o="342v" n="685"/>
<pb file="add_6782_f343" o="343" n="686"/>
<pb file="add_6782_f343v" o="343v" n="687"/>
<pb file="add_6782_f344" o="344" n="688"/>
<pb file="add_6782_f344v" o="344v" n="689"/>
<pb file="add_6782_f345" o="345" n="690"/>
<pb file="add_6782_f345v" o="345v" n="691"/>
<pb file="add_6782_f346" o="346" n="692"/>
<pb file="add_6782_f346v" o="346v" n="693"/>
<pb file="add_6782_f347" o="347" n="694"/>
<div xml:id="echoid-div226" type="page_commentary" level="2" n="226">
<p>
<s xml:id="echoid-s870" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s870" xml:space="preserve">
The table from page 12 of the 'Magisteria' (Add MS 6782, f. 119).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f347v" o="347v" n="695"/>
<pb file="add_6782_f348" o="348" n="696"/>
<div xml:id="echoid-div227" type="page_commentary" level="2" n="227">
<p>
<s xml:id="echoid-s872" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s872" xml:space="preserve">
See page 5 of the 'Magisteria' (Add MS 6782, f. 112), which contains the same numerical tables.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f348v" o="348v" n="697"/>
<div xml:id="echoid-div228" type="page_commentary" level="2" n="228">
<p>
<s xml:id="echoid-s874" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s874" xml:space="preserve">
See page 7 of the 'Magisteria' (Add MS 6782, f. 114), which contains the first two numerical tables.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f349" o="349" n="698"/>
<div xml:id="echoid-div229" type="page_commentary" level="2" n="229">
<p>
<s xml:id="echoid-s876" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s876" xml:space="preserve">
Formulae for entries in the <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>c</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> columns of a difference table;
see page 16 of the 'Magisteria' (Add MS 6782, f. 123).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f349v" o="349v" n="699"/>
<pb file="add_6782_f350" o="350" n="700"/>
<pb file="add_6782_f350v" o="350v" n="701"/>
<pb file="add_6782_f351" o="351" n="702"/>
<pb file="add_6782_f351v" o="351v" n="703"/>
<pb file="add_6782_f352" o="352" n="704"/>
<pb file="add_6782_f352v" o="352v" n="705"/>
<pb file="add_6782_f353" o="353" n="706"/>
<pb file="add_6782_f353v" o="353v" n="707"/>
<pb file="add_6782_f354" o="354" n="708"/>
<div xml:id="echoid-div230" type="page_commentary" level="2" n="230">
<p>
<s xml:id="echoid-s878" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s878" xml:space="preserve">
Squares of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo maxsize="1">)</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>+</mo><mi>f</mi><mo maxsize="1">)</mo></mstyle></math>.
The page also shows the calculation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>b</mi><mo>-</mo><mi>c</mi><mo maxsize="1">)</mo></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head127" xml:space="preserve" xml:lang="lat">
Quadrata e polynomia radice
<lb/>[<emph style="it">tr:
Squares from polynomial roots.
</emph>]<lb/>
</head>
<pb file="add_6782_f354v" o="354v" n="709"/>
<pb file="add_6782_f355" o="355" n="710"/>
<div xml:id="echoid-div231" type="page_commentary" level="2" n="231">
<p>
<s xml:id="echoid-s880" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s880" xml:space="preserve">
Squares of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>–</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>–</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>–</mo><mi>c</mi><mo>–</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>,
and of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>–</mo><mi>f</mi><mo maxsize="1">)</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>–</mo><mi>d</mi><mo>–</mo><mi>f</mi><mo maxsize="1">)</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>–</mo><mi>c</mi><mo>–</mo><mi>d</mi><mo>–</mo><mi>f</mi><mo maxsize="1">)</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>–</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>–</mo><mi>f</mi><mo maxsize="1">)</mo></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head128" xml:space="preserve" xml:lang="lat">
Quadrata e polynomia radice
<lb/>[<emph style="it">tr:
Squares from polynomial roots.
</emph>]<lb/>
</head>
<pb file="add_6782_f355v" o="355v" n="711"/>
<pb file="add_6782_f356" o="356" n="712"/>
<div xml:id="echoid-div232" type="page_commentary" level="2" n="232">
<p>
<s xml:id="echoid-s882" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s882" xml:space="preserve">
The square and cube of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f356v" o="356v" n="713"/>
<pb file="add_6782_f357" o="357" n="714"/>
<div xml:id="echoid-div233" type="page_commentary" level="2" n="233">
<p>
<s xml:id="echoid-s884" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s884" xml:space="preserve">
Powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo maxsize="1">)</mo></mstyle></math> up to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mrow><msup><mo maxsize="1">)</mo><mn>6</mn></msup></mrow></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head129" xml:space="preserve" xml:lang="lat">
potentia e binomia radice
<lb/>[<emph style="it">tr:
powers from binomial roots
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s886" xml:space="preserve">
solidum
<lb/>[<emph style="it">tr:
solid
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f357v" o="357v" n="715"/>
<pb file="add_6782_f358" o="358" n="716"/>
<div xml:id="echoid-div234" type="page_commentary" level="2" n="234">
<p>
<s xml:id="echoid-s887" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s887" xml:space="preserve">
Cubes of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>+</mo><mi>f</mi><mo maxsize="1">)</mo></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>+</mo><mi>f</mi><mo>+</mo><mi>g</mi><mo maxsize="1">)</mo></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6782_f358v" o="358v" n="717"/>
<pb file="add_6782_f359" o="359" n="718"/>
<pb file="add_6782_f359v" o="359v" n="719"/>
<pb file="add_6782_f360" o="360" n="720"/>
<pb file="add_6782_f360v" o="360v" n="721"/>
<pb file="add_6782_f361" o="361" n="722"/>
<pb file="add_6782_f361v" o="361v" n="723"/>
<pb file="add_6782_f362" o="362" n="724"/>
<div xml:id="echoid-div235" type="page_commentary" level="2" n="235">
<p>
<s xml:id="echoid-s889" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s889" xml:space="preserve">
This page refers to Aristotle's <emph style="it">Physics</emph>, Books V and VI,
defines what it means for things to be together or apart, in contact, or continuous.
The definitions may be paraphrased as follows.
</s>
<lb/>
<quote>
Things are said to be together if they are in one place, apart if they are in different places. <lb/>
Things are said to be in contact if their extremities are together. <lb/>
Things are said to be continuous if the touching limits of each become one and the same.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head130" xml:space="preserve" xml:lang="lat">
De infinitis.	De continuo.
<lb/>[<emph style="it">tr:
On infinity. On the continuum.
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s891" xml:space="preserve">
Aristotle in the beginning of his 6th booke of his physicks, &amp; in the <lb/>
26th treatise of the 5th booke, defineth those thinges to be <foreign xml:lang="lat">continua <lb/>
quorum extrema sunt unum.</foreign> And in the 22nd treatise of the said 5th booke <lb/>
that: <foreign xml:lang="lat">tangentia sunt, quorum extrema sunt simul</foreign>.
<foreign xml:lang="lat">Simul qua in <lb/>
uno loco sunt primo</foreign>. <foreign xml:lang="lat">Separatim qui sunt in altero.</foreign>
</s>
</p>
<p>
<s xml:id="echoid-s892" xml:space="preserve">
Now for the <emph style="st">understanding</emph>
<emph style="super">better explication</emph> of the
<emph style="super">meaning of the</emph> definitions as also of their truth. Let us <lb/>
understand first two <emph style="super">materiall</emph>
cubes A &amp; B to be separate, that is, to be in diverse <lb/>
planes, extremes &amp; all.
</s>
</p>
<pb file="add_6782_f362v" o="362v" n="725"/>
<pb file="add_6782_f363" o="363" n="726"/>
<head xml:id="echoid-head131" xml:space="preserve" xml:lang="lat">
De Infinitis progressionibus
<lb/>[<emph style="it">tr:
On infinite progressions
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s893" xml:space="preserve">
In progressions that be infinite be they increasing or decreasing. <lb/>
</s>
<s xml:id="echoid-s894" xml:space="preserve">
There are these passes.
</s>
<s xml:id="echoid-s895" xml:space="preserve">
First to a quantity that haveth no
<emph style="st">proportion</emph> rate <lb/>
to the first quantity given, or rather because betwixt positive quantityes <lb/>
there is a positive rate, I may call that rate infinite either in great-<lb/>
ness or litle<emph style="super">nes</emph>s according to the
<emph style="st">proportion</emph> <emph style="super">progression</emph>,
in respect of the first quantity <lb/>
given.
</s>
<s xml:id="echoid-s896" xml:space="preserve">
Yet in respecte of the progression following it is divisible or mul-<lb/>
tiplicable till the progression being infinite hath for his second passe <lb/>
also a quantity <emph style="super">of an[???]</emph> infinite rate.
</s>
<s xml:id="echoid-s897" xml:space="preserve">
Which is not only infinite in respecte of <lb/>
the first quantity of the last progression; but infinitely infinite in respect <lb/>
of <emph style="st">of</emph> the first in the first progresse.
</s>
<s xml:id="echoid-s898" xml:space="preserve">
And also the summe of the second pro-<lb/>
gression is infinite <emph style="st">infi</emph>
in respect of the first summe of the first pro-<lb/>
gression, or the first quantity of all.
</s>
</p>
<p>
<s xml:id="echoid-s899" xml:space="preserve">
And so a third, fourth &amp; infinite other progressions and passes; of which <lb/>
any quantity or the summe of all infinitely all, is of an infinite <lb/>
quantity in greatness of litleness in respect, of the summe or <lb/>
first quantity of the first progression.
</s>
<lb/>
<s xml:id="echoid-s900" xml:space="preserve">
And yet <emph style="st">at</emph> <emph style="super">for a</emph>
last in decreasing progressions we must needes under-<lb/>
stand a quantity absolutely indivisible; but multiplicable infinitely <lb/>
infinite <emph style="st">to make the [¿]prime[?]
from where the rest are issued</emph> till a quantity <lb/>
absolutely immultiplicable be produced which I may call universally infinite.
</s>
<lb/>
<s xml:id="echoid-s901" xml:space="preserve">
And in increasing progressions we must needes understand that <lb/>
<emph style="st">at</emph> <emph style="super">for a</emph>
last there must be a quantity immultiplicable absolute, but <lb/>
divisible infinitely infinite till that quantity be issued that is <lb/>
absolutely indivisble.
</s>
</p>
<p>
<s xml:id="echoid-s902" xml:space="preserve">
That such <emph style="st">a</emph> quantity which I call universally infinite: hath not only <lb/>
act rationall, by supposition, or by consequence from
<emph style="super">mere</emph> supposition: but <lb/>
also act reall, or existence: in an instant, having
<emph style="super">[???] perfect</emph> actuall being, <lb/>
or in time, passed by motion <emph style="st">fini</emph>
both finite &amp; infinite: with many reall <lb/>
consequences or properties consequent; &amp; accidents adioyning: <lb/>
shalbe declared in the papers following.
</s>
</p>
<pb file="add_6782_f363v" o="363v" n="727"/>
<pb file="add_6782_f364" o="364" n="728"/>
<head xml:id="echoid-head132" xml:space="preserve" xml:lang="lat">
De Infinitis.
<lb/>[<emph style="it">tr:
On infinity
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s903" xml:space="preserve">
Seing that any finite line will <lb/>
subtend an angle at summe distance; <lb/>
as let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> subtend the <emph style="st">the</emph> angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi><mi>c</mi></mstyle></math>.
</s>
<lb/>
<s xml:id="echoid-s904" xml:space="preserve">
Then a line double to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, which let be <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math>, will subtend the same angle at a <lb/>
double distance, so that <emph style="it"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math></emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi></mstyle></math> will be <lb/>
aequall to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>.
</s>
</p>
<p>
<s xml:id="echoid-s905" xml:space="preserve">
In those subtensions I understand that the poynt <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be <emph style="super">in a</emph>
perpendicular <emph style="super">line</emph> to the <lb/>
middle of the subtendent lines.
</s>
<s xml:id="echoid-s906" xml:space="preserve">
as also in all the others which follow.
</s>
</p>
<p>
<s xml:id="echoid-s907" xml:space="preserve">
Now I suppose <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> to be removed to a further distance from the poynt <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/>
</s>
<s xml:id="echoid-s908" xml:space="preserve">
Then the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi><mi>c</mi></mstyle></math> subtended must be lesse than before.
</s>
<s xml:id="echoid-s909" xml:space="preserve">
And <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math>. <lb/>
shall <emph style="st">[???]</emph> subtend the same angle at a double distance as before.
</s>
</p>
<p>
<s xml:id="echoid-s910" xml:space="preserve">
And this is true <emph style="st">generally</emph> continually that the further
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> is removed <lb/>
the lesse angle it subtendeth &amp; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math> always must subtend the same <lb/>
angle at a double distance.
</s>
</p>
<p>
<s xml:id="echoid-s911" xml:space="preserve">
Then I suppose <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> to be removed to an infinite distance; at which <lb/>
distance the supposition altereth not the quantity of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. but the
<emph style="st">quantity </emph>consequence <lb/>
is of the angle.
</s>
<s xml:id="echoid-s912" xml:space="preserve">
Which wilbe, that the angle <emph style="st">wh</emph>
then subtended <emph style="it">[???]</emph> to be <lb/>
of an infinite quantity in litleness in respecte of the former angles.
</s>
<s xml:id="echoid-s913" xml:space="preserve">
Yet it <lb/>
cannot be sayd to be no angle negatively because it is positive. &amp; it <lb/>
must also follow that the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math> must subtend the same positive angle <lb/>
at a double distance.
</s>
<s xml:id="echoid-s914" xml:space="preserve">
Which is Double to the former infinite distance.
</s>
</p>
<p>
<s xml:id="echoid-s915" xml:space="preserve">
Also, let the distance of the subtendents be nearer <emph style="st">[???]</emph>
<emph style="super">to</emph> infinite,
<emph style="st">[???]</emph>it cannot be <lb/>
otherwise inferred but that the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math> &amp; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>g</mi></mstyle></math>
<emph style="st">being infinit</emph> though infinite, <lb/>
be <foreign xml:lang="lat ">ad diversas partes, &amp; in diversis locis</foreign>,
because <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> &amp; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math> are betweene them, <lb/>
&amp; have agreement or concurrence but only in the poynt <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>,
<emph style="st">[???]</emph> <emph style="super">or</emph> in no distance <lb/>
out of the poynt <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>.
</s>
</p>
<p>
<s xml:id="echoid-s916" xml:space="preserve">
And yet the nearness of there congruence &amp;
con<emph style="super">cu</emph>rrence in all other partes <lb/>
[???] at the utmost is such, that although they be remote; the angle <lb/>
is of no proportion explicable by nomber finite, but infinite
[¿]unknown[?], to any <lb/>
<emph style="st">angles</emph> other angle which we call finite.
</s>
<s xml:id="echoid-s917" xml:space="preserve">
The like inexplicable proportion <lb/>
is of the <emph style="super">subtendent</emph>
lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math> &amp; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, to there infinite distance position from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>.
</s>
<lb/>
<s xml:id="echoid-s918" xml:space="preserve">
And yet the sayd lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math> &amp; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. as also that infinite litle or improportio-<lb/>
nable angle is divisible still <foreign xml:lang="lat">in infinitum</foreign>. &amp;
still, although improportionable <lb/>
yet in an other respect, that is to say of his owne partes, is proportionable.
</s>
</p>
<pb file="add_6782_f364v" o="364v" n="729"/>
<pb file="add_6782_f365" o="365" n="730"/>
<head xml:id="echoid-head133" xml:space="preserve" xml:lang="lat">
De Infinitis.
<lb/>[<emph style="it">tr:
On infinity
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s919" xml:space="preserve">
That in a finite time an infinite space <lb/>
may be moved
</s>
</p>
<p>
<s xml:id="echoid-s920" xml:space="preserve">
It is now convenient <lb/>
that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>f</mi></mstyle></math> be in this line.
</s>
</p>
<p>
<s xml:id="echoid-s921" xml:space="preserve">
Suppose the <emph style="st">the</emph> line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>e</mi><mi>f</mi></mstyle></math> <foreign xml:lang="lat">et ultra</foreign> <lb/>
to be infinite, &amp; the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> suppose <lb/>
to revolve &amp; describe a circle <lb/>
in a finite time, fro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> towards <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> doth first respect <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <lb/>
after <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, &amp; so forth successively no poynt <lb/>
in the infinite line is <emph style="st">not</emph>
unrespected by that time the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> cometh <lb/>
to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>g</mi></mstyle></math> where then the line is parallel &amp; cutteth not the former <lb/>
line infinite.
</s>
<s xml:id="echoid-s922" xml:space="preserve">
Now seing that a motion may be of any thing <lb/>
according <emph style="super">to</emph> the continuall succession of a poynt,
as well in respect <lb/>
of <emph style="st">[???]</emph>
<foreign xml:lang="lat">mobile ab motus</foreign>.
</s>
<s xml:id="echoid-s923" xml:space="preserve">
Whatsoever may be or not be in <lb/>
respect of the moment, it maketh no matter: the purpose is <lb/>
manifest.
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s924" xml:space="preserve">
Consequentia <lb/>
Accidentis quædam huius motus.
</s>
</p>
<p>
<s xml:id="echoid-s925" xml:space="preserve">
The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> having moved till he comes to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math> that is <lb/>
parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>f</mi></mstyle></math>. &amp; so that continuing his motion of revolution:
</s>
</p>
<p>
<s xml:id="echoid-s926" xml:space="preserve">
The lines are parallel but in one instant.
</s>
<lb/>
<s xml:id="echoid-s927" xml:space="preserve">
They never cut at an infinite distance but at that instant <lb/>
they are parallel.
</s>
<lb/>
<s xml:id="echoid-s928" xml:space="preserve">
And if they cut then, they must cut
<foreign xml:lang="lat">ad utrasque partes</foreign> &amp; then <lb/>
being right lines there must be no space betwixte them, but <lb/>
there distance by supposition is more than the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>.
</s>
<s xml:id="echoid-s929" xml:space="preserve">
Which <lb/>
implies contradiction.
</s>
</p>
<p>
<s xml:id="echoid-s930" xml:space="preserve">
And yet there must be a cutting at an infinite distance or else all <lb/>
the poyntes of the infinite line could not have been respected. &amp;
</s>
<s xml:id="echoid-s931" xml:space="preserve">
if <lb/>
that be not some part of the infinite line,
that is some quantity <emph style="st">which</emph> <lb/>
<emph style="st">it is</emph> finite is only cut;
&amp; that is at a finite distance; &amp; then it maketh an <lb/>
angle <emph style="super">of quantity</emph>
at the greatest distance of such cutting: from that cutting the line <lb/>
by motion came to be parallel: That motion is made in an instant or <lb/>
in time.
</s>
<s xml:id="echoid-s932" xml:space="preserve">
If in time, then in half the time the cutting must be further <lb/>
than the supposed furthest;
</s>
<s xml:id="echoid-s933" xml:space="preserve">
If in an instant, our line wilbe in <emph style="st">two places</emph> <lb/>
two places in one <emph style="st">[???]</emph> instant;
<foreign xml:lang="lat">quæ implicant</foreign>.
</s>
</p>
<p>
<s xml:id="echoid-s934" xml:space="preserve">
The lines therefore must cut at an infinite distance before they come to <lb/>
be parallel.
</s>
<s xml:id="echoid-s935" xml:space="preserve">
And that must be in time before or in an instant before. <lb/>
</s>
<s xml:id="echoid-s936" xml:space="preserve">
If in time, then in half the time they cut at greater distance than infinite or <lb/>
are parallel before they are parallel.
</s>
<s xml:id="echoid-s937" xml:space="preserve">
Which both do imply contradiction. <lb/>
</s>
<s xml:id="echoid-s938" xml:space="preserve">
If in an instant before; the two instants are one or different.
</s>
<s xml:id="echoid-s939" xml:space="preserve">
If one, <foreign xml:lang="lat">implicat</foreign>. <lb/>
</s>
<s xml:id="echoid-s940" xml:space="preserve">
If two there must be no other betwixt them.
</s>
<s xml:id="echoid-s941" xml:space="preserve">
And then there [???] be a time greater <lb/>
than an instant &amp; lesse than any time of quantity that is indivisible, that is <lb/>
agayne, indivisible into partes of quantity. &amp; so also like of poyntes &amp;c.
</s>
</p>
<pb file="add_6782_f365v" o="365v" n="731"/>
<pb file="add_6782_f366" o="366" n="732"/>
<head xml:id="echoid-head134" xml:space="preserve" xml:lang="lat">
De Infinitis.
<lb/>[<emph style="it">tr:
On infinity
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s942" xml:space="preserve">
<emph style="st">If the</emph> <emph style="super">The</emph> line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> by his revolution <lb/>
cometh at length to be parallel to <lb/>
the infinite line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math>.
</s>
<s xml:id="echoid-s943" xml:space="preserve">
Which <lb/>
motion being from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> suppose <lb/>
to have been æqually.
</s>
<s xml:id="echoid-s944" xml:space="preserve">
The <lb/>
degree of the motion let be <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>n</mi></mstyle></math>. the time <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>p</mi></mstyle></math>.
</s>
<s xml:id="echoid-s945" xml:space="preserve">
The beginning of the time or first instant <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. <lb/>
</s>
<s xml:id="echoid-s946" xml:space="preserve">
The last instant wherein the line is <lb/>
parallel, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. Now seing that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> must cut at <lb/>
an infinite distance &amp; <emph style="super">that</emph> his last cutting must be <lb/>
before the instant <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>.
</s>
<s xml:id="echoid-s947" xml:space="preserve">
Which suppose <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math>.
</s>
<s xml:id="echoid-s948" xml:space="preserve">
That <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math> as it is argued by the premises <lb/>
must differe from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> by an indivisible time, so that <emph style="st">it </emph>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math> must be the next instant <lb/>
to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. &amp; no other between.
</s>
<s xml:id="echoid-s949" xml:space="preserve">
In which instant <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> must not be parallel but <lb/>
make his last cutting at an infinite distance.
</s>
<s xml:id="echoid-s950" xml:space="preserve">
And therefore it must have <lb/>
a certayne <foreign xml:lang="lat">situs</foreign>
at that instant out of the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> towards <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, which let be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math>, <lb/>
as it maketh his last section.
</s>
<s xml:id="echoid-s951" xml:space="preserve">
In which situation the motion ordering it hath <lb/>
the sayd degree <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>n</mi></mstyle></math>, as in all other situations.
</s>
<s xml:id="echoid-s952" xml:space="preserve">
From the which situation to the situation <lb/>
of being parallel it must be moved unto (as it is sayd) in the next instant.
</s>
</p>
<p>
<s xml:id="echoid-s953" xml:space="preserve">
Now suppose (as it may be) that the motion from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> be in half the time <lb/>
of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>p</mi></mstyle></math>.
</s>
<s xml:id="echoid-s954" xml:space="preserve">
Then doth it follow necessarily that the degree of motion or velo-<lb/>
city be double to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>n</mi></mstyle></math>. And therefore, what space or parte of a space, (be it <lb/>
finite or infinite, so it be positive,) it moved before according to <lb/>
the degree of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>n</mi></mstyle></math>. it moveth the same now, in half the time. <lb/>
</s>
<s xml:id="echoid-s955" xml:space="preserve">
Therefore in this second motion when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> cometh to have his situation <lb/>
at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math> to make the sayd last section; seing that then it hath double <lb/>
degree of velocity; it must afterward be parallel in half an instant <lb/>
that is to say, <emph style="st">that</emph>
in half that time which was sayd to be indivisible. <lb/>
</s>
<s xml:id="echoid-s956" xml:space="preserve">
Which doth imply contradiction.
</s>
</p>
<p>
<s xml:id="echoid-s957" xml:space="preserve">
Agayne if it be sayd that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math> at that instant
<emph style="st">&amp; in the position</emph> (when &amp; <lb/>
where it maketh his last section with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> before it be parallel)
<emph style="st">then</emph> <lb/>
be <foreign xml:lang="lat">deinceps</foreign> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>. or that the poynts <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> &amp; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> be
<foreign xml:lang="lat">deinceps</foreign> at an infinite <lb/>
distance so that no point can be between.
</s>
<s xml:id="echoid-s958" xml:space="preserve">
Yet from the poynt <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> may <lb/>
be interposed a line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>f</mi></mstyle></math>. and also from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. &amp; by the doctrine of Elements <lb/>
the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>k</mi><mi>h</mi></mstyle></math>, or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>l</mi><mi>h</mi></mstyle></math> must be
<emph style="st">greater</emph> <emph style="super">lesser</emph>
than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>a</mi><mi>h</mi></mstyle></math>. &amp; therefore lesse than that <lb/>
which was sayd to be least or indivisible. &amp; therefore the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math> &amp; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>, or the <lb/>
poynts <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> &amp; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> be not <foreign xml:lang="lat">deinceps. quæ implicant</foreign>.
</s>
</p>
<pb file="add_6782_f366v" o="366v" n="733"/>
<pb file="add_6782_f367" o="367" n="734"/>
<head xml:id="echoid-head135" xml:space="preserve" xml:lang="lat">
De Infinitis.	Ratio Achilles
<lb/>[<emph style="it">tr:
On infinity. The ratio of Achilles
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s959" xml:space="preserve">
There is a reason <emph style="super">of Zeno</emph> in Aristotle
(in the 6th booke of his phisickes. text. 78.) which <lb/>
for the sorce it seemeth to carry is called Achilles.
</s>
<s xml:id="echoid-s960" xml:space="preserve">
And for that cause, no doubt, <lb/>
is <emph style="super">the</emph> name also Achilles used in the example to expresse the reason.
</s>
<s xml:id="echoid-s961" xml:space="preserve">
The which <lb/>
because it is against Aristotles doctrine
&amp; for that it compryseth matter <emph style="super">pregnant</emph> <lb/>
of greater consequence concerning the doctrine of infinites, it being there <lb/>
but briefly &amp; obscurely set downe with an answere uncertayne: I thinke good <lb/>
to set <emph style="st">[???]</emph>downe more [???] &amp; largely:
with Aristotles Answere as he hath <lb/>
it in the place allwayes, as also at full according to his owne doctrine in <lb/>
other places.
</s>
<s xml:id="echoid-s962" xml:space="preserve">
To the end that comparing one with the other, the truth may appear, <lb/>
&amp; perhaps [¿]seem[?] otherwise to be,
then yet hath been by the peripateticles either noted or <lb/>
observed.
</s>
</p>
<p>
<s xml:id="echoid-s963" xml:space="preserve">
The proposition of Zeno is.
</s>
<s xml:id="echoid-s964" xml:space="preserve">
The swift runner (runne he never so <lb/>
swiftly) shall never overtake the slow runner <emph style="super">mover</emph>
(runne <emph style="super">move</emph> he never <lb/>
so slowly.
</s>
<lb/>
<s xml:id="echoid-s965" xml:space="preserve">
That there may be <emph style="super">no</emph> doubte of the meaning of the <lb/>
proposition we will declare what thinges are therein supposed.
</s>
<lb/>
<s xml:id="echoid-s966" xml:space="preserve">
<emph style="st">The suppositions for the reason are adjoyned.</emph>
</s>
<lb/>
<s xml:id="echoid-s967" xml:space="preserve">
ffirst, (as it ought to be, else the proposition were ridiculous) The motion <lb/>
of the runner &amp; slow mover are understood to be both one way &amp; in <lb/>
one right line.
</s>
<lb/>
<s xml:id="echoid-s968" xml:space="preserve">
Secondly <emph style="st">the [???] of [???] must be of some [???]</emph>
The <lb/>
</s>
</p>
<pb file="add_6782_f367v" o="367v" n="735"/>
<pb file="add_6782_f368" o="368" n="736"/>
<head xml:id="echoid-head136" xml:space="preserve" xml:lang="lat">
Ratio Achilles.
<lb/>[<emph style="it">tr:
The ratio of Achilles
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s969" xml:space="preserve">
Let Achilles be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>.
</s>
<lb/>
<s xml:id="echoid-s970" xml:space="preserve">
Testudo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
The tortoise <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>.
</emph>]<lb/>
</s>
<s xml:id="echoid-s971" xml:space="preserve">
The Motion of Achilles from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> <lb/>
in the time <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math>. of Testudo from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math> <lb/>
in the <emph style="super">same</emph> time <emph style="st">fg</emph>.
</s>
<s xml:id="echoid-s972" xml:space="preserve">
<emph style="st">Which let be the</emph> <lb/>
<emph style="st">half parte of the time ef</emph>.
</s>
<s xml:id="echoid-s973" xml:space="preserve">
Which <lb/>
space of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math> let be the tenth parte <lb/>
of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>.
</s>
<s xml:id="echoid-s974" xml:space="preserve">
Now the quaestion is, <lb/>
both these motions being continued in the same proportion as 10 to 1. <lb/>
where &amp; when shall <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> overtake <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>. <emph style="st">Suppose at d</emph>. <lb/>
</s>
<s xml:id="echoid-s975" xml:space="preserve">
At some point or other it must really be.
</s>
<s xml:id="echoid-s976" xml:space="preserve">
Suppose that <emph style="super">X</emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
</s>
<s xml:id="echoid-s977" xml:space="preserve">
There must be <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> &amp; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>, <emph style="st">[???]</emph>
at the same instant of time.
</s>
<s xml:id="echoid-s978" xml:space="preserve">
And therefore the time wherein <lb/>
<emph style="st">that</emph>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> hath moved to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> must be the same wherein <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> hath moved to <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
</s>
<s xml:id="echoid-s979" xml:space="preserve">
But the space <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>d</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>d</mi></mstyle></math> must be as 10 to 1.
</s>
<lb/>
<s xml:id="echoid-s980" xml:space="preserve">
Now by the supposition it must follow
(because these motions be proportionall <emph style="super">(as 10 to 1)</emph>) <lb/>
* As <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math>. so: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>d</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>d</mi></mstyle></math>. which same termes <lb/>
proportionall call by these <emph style="st">same</emph> letters &amp; in the same order.
</s>
</p>
<p>
<s xml:id="echoid-s981" xml:space="preserve">
As <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math> is known to be 1. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>γ</mi></mstyle></math> is <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>.
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mo>+</mo><mi>α</mi></mstyle></math> is unknown. &amp; so is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi></mstyle></math>. <lb/>
yet this is known that. <emph style="st">is æquall to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ɛ</mi></mstyle></math>.</emph>
</s>
</p>
<p>
<s xml:id="echoid-s982" xml:space="preserve">
* Now what other proportion is this than if a man <lb/>
should say as <emph style="st">all</emph> the first to the second so <lb/>
all the antecedents to all the consequents which <lb/>
in this be infinite in nomber.
</s>
</p>
<p>
<s xml:id="echoid-s983" xml:space="preserve">
X To find that poynt geometrically is set downe <lb/>
in my other papers <foreign xml:lang="lat">de infinitis</foreign>.
<lb/>[<emph style="it">tr:
on infinity
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f368v" o="368v" n="737"/>
<pb file="add_6782_f369" o="369" n="738"/>
<head xml:id="echoid-head137" xml:space="preserve" xml:lang="lat">
De Infinitis.
<lb/>[<emph style="it">tr:
On infinity
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s984" xml:space="preserve">
Now will I propound some dfficultyes to be <lb/>
considered of.
</s>
<s xml:id="echoid-s985" xml:space="preserve">
Seing that every line is compounded <lb/>
of atomes, &amp; therefore the periphery of a circle. <emph style="st">that <lb/>
is to say</emph> one
<foreign xml:lang="lat">atomus</foreign> is succeeding one an other <lb/>
infinitely in such manner as <emph style="it">that</emph> the perifery is at <lb/>
last compounded and made.
</s>
</p>
<p>
<s xml:id="echoid-s986" xml:space="preserve">
Now also seing that the whole <foreign xml:lang="lat">periferies</foreign>
is compounded of <foreign xml:lang="lat">atomis undiquaque <lb/>
sitis</foreign> about the poynt <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. so many times infinitely, &amp; to that number of them <lb/>
infinitely, till the circle supposed be accomplished.
</s>
</p>
<p>
<s xml:id="echoid-s987" xml:space="preserve">
I demand <emph style="st">therefore</emph> <emph style="super">then</emph>
what wilbe the nomber of <foreign xml:lang="lat">atomi</foreign>
that are <foreign xml:lang="lat">deinceps</foreign> about the <lb/>
point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>.
</s>
<s xml:id="echoid-s988" xml:space="preserve">
Infinite they must needes be, or else infinite lines could not <lb/>
be <emph style="st">dra</emph> supposed actually from the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> to the perifery.
</s>
<s xml:id="echoid-s989" xml:space="preserve">
And infinite also <lb/>
are <emph style="super">also</emph> in the perifery.
</s>
<s xml:id="echoid-s990" xml:space="preserve">
But now I demande whether they are aequally infinite <lb/>
or not.
</s>
<s xml:id="echoid-s991" xml:space="preserve">
If about the center are lesse infinite then there cannot from the <lb/>
center <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> to every poynt in the perifery be understood a right line but <lb/>
we must understand those <emph style="super"><foreign xml:lang="lat">atomi</foreign> about the center</emph>
that we supposed indivisible, divisble <emph style="super">which were absurd</emph>.
</s>
<s xml:id="echoid-s992" xml:space="preserve">
and <lb/>
if they be æqually infinite: then <emph style="st">the same nomber of
<foreign xml:lang="lat">atomi</foreign></emph> in a great <lb/>
place, (where the nomber, although infinite, yet in them selves definite; because <lb/>
they being supposed to have <emph style="st">[???]</emph>
acte there is not one more nor lesse.
</s>
<s xml:id="echoid-s993" xml:space="preserve">
Neither <lb/>
can there be more because <emph style="st">[???]</emph>
they being <foreign xml:lang="lat">deinceps</foreign> one more cannot <lb/>
be between there being no distance: &amp; if there <emph style="st">be supposed</emph>
<emph style="super">might be</emph> one lesse; there <lb/>
lacketh of the supposed actaull, &amp; definite &amp; positive number although infinite. <lb/>
</s>
<s xml:id="echoid-s994" xml:space="preserve">
Then I say in a greate place where there could be no more or lesse, <lb/>
in a lesse place there are an æquall nomber; which seemeth to imply.
</s>
</p>
<p>
<s xml:id="echoid-s995" xml:space="preserve">
An other difficulty riseth from the square.
</s>
<s xml:id="echoid-s996" xml:space="preserve">
If a line <lb/>
be compounded of <foreign xml:lang="lat">atomis</foreign>, the diametrall line wilbe <lb/>
found to be aæquall to the side.
</s>
<s xml:id="echoid-s997" xml:space="preserve">
ffor suppose the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> <lb/>
to be drawne <emph style="st">from the poynt[???]</emph>
from the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> <emph style="super">of the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math>,</emph> to <lb/>
the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, of <emph style="st">th</emph> the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. Then from the next point <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, which is <foreign xml:lang="lat">deinceps</foreign>
to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> in the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math>, draw a line <lb/>
to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> the next point to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> in the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>.
</s>
<s xml:id="echoid-s998" xml:space="preserve">
So likewise from every <lb/>
next point in the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math>, to every next point in the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. <lb/>
</s>
<s xml:id="echoid-s999" xml:space="preserve">
Now the lines so drawne must needs be the least &amp; most that may be, <lb/>
because they are <foreign xml:lang="lat">deinceps</foreign> &amp; all.
&amp; they all cut the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> &amp; of the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> <lb/>
there can be <emph style="st">between</emph> no point betwixt
<emph style="st">the</emph> two of the former lines
<emph style="super">because they are
<foreign xml:lang="lat">deinceps</foreign></emph>.
</s>
<s xml:id="echoid-s1000" xml:space="preserve">
And therefore <lb/>
the nomber of the poynts of the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>,
<emph style="super">are</emph> aequally infinite to the poynts of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> &amp; <lb/>
<foreign xml:lang="lat">per</foreign> consequence the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> &amp; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> aequall.
</s>
<s xml:id="echoid-s1001" xml:space="preserve">
But this difficulty wilbe made more <lb/>
playne by the next following, which <emph style="st">[???]</emph>
wilbe found the meanes for the solution <lb/>
of all.
</s>
</p>
<p>
<s xml:id="echoid-s1002" xml:space="preserve">
An other question is. where two
<foreign xml:lang="lat">atomi</foreign> are
<foreign xml:lang="lat">deinceps</foreign>. whether an other <lb/>
(the <emph style="st">other</emph> <emph style="super">first two</emph>
not disioyned) may either passe or have situation betwixt them.
</s>
</p>
<pb file="add_6782_f369v" o="369v" n="739"/>
<pb file="add_6782_f370" o="370" n="740"/>
<head xml:id="echoid-head138" xml:space="preserve" xml:lang="lat">
De Infinitis. Notanda.
<lb/>[<emph style="it">tr:
On infinity. To be noted.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1003" xml:space="preserve">
De tactu duorum corporum <lb/>
per superficies. an duæ <lb/>
superficies sint realiter distantes <lb/>
in corporum contactu.
<lb/>[<emph style="it">tr:
On the contact of two bodies at their surfaces,
but the two surfaces are in reality separate in the contact of the bodies.
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s1004" xml:space="preserve">
Because <foreign xml:lang="lat">discretum</foreign>
is negative to <foreign xml:lang="lat">continuum</foreign> <lb/>
in respect of that thing <emph style="st">that</emph>
<emph style="super">which</emph> may be sayd to be <lb/>
either.
</s>
<s xml:id="echoid-s1005" xml:space="preserve">
If yet <emph style="super">that</emph>
which is <foreign xml:lang="lat">discretum</foreign>
is not <foreign xml:lang="lat">continuum</foreign> <lb/>
&amp; that which is <foreign xml:lang="lat">continuum</foreign>
is not <foreign xml:lang="lat">discretum</foreign>. therefore <lb/>
the one being knowne the other cannot be <lb/>
unknowne what it is.
</s>
<s xml:id="echoid-s1006" xml:space="preserve">
Now although there <lb/>
be great controversy of the essence &amp; quality <lb/>
of <foreign xml:lang="lat">continuum</foreign>.
yet there is no such of <foreign xml:lang="lat">discretum</foreign>. <lb/>
we will therefore lay downe what is manifest <lb/>
of it, that the ratio &amp; essence of <foreign xml:lang="lat">continuum</foreign> may appeare.
</s>
</p>
<pb file="add_6782_f370v" o="370v" n="741"/>
<p>
<s xml:id="echoid-s1007" xml:space="preserve">
Willaim Sprat a wolle draper <lb/>
at the sign of the rope in Watlin <lb/>
street at Soper Lane corner. serveth <lb/>
for a [???] for his wifes brother <lb/>
for [???]. there are 4.
</s>
</p>
<pb file="add_6782_f371" o="371" n="742"/>
<head xml:id="echoid-head139" xml:space="preserve" xml:lang="lat">
De Infinitis.
<lb/>[<emph style="it">tr:
On infinity
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s1008" xml:space="preserve">
That there may be two magnitudes given, of which the one shalbe <lb/>
infinite in respect of the other, &amp; yet in respect of two other <lb/>
magnitudes they shalbe finite.
</s>
</p>
<p>
<s xml:id="echoid-s1009" xml:space="preserve">
That a line finite, cannot have his partes, of a finite magnitude; but <lb/>
they must be of a finite nomber.
</s>
</p>
<p>
<s xml:id="echoid-s1010" xml:space="preserve">
That a finite line may have an infinite nomber of partes,
&amp; if <emph style="super">all</emph> the <lb/>
partes be in continuall proportion: the nomber must be compounded <lb/>
of an infinite nomber of finite partes; &amp; an infinite nomber of <lb/>
infinite partes.
</s>
</p>
<p>
<s xml:id="echoid-s1011" xml:space="preserve">
If a line be understood to be compounded of
<emph style="st">infinite</emph> poyntes: <lb/>
the nomber of them is infinite of the first passe, second or <lb/>
any nomber of passes finite or infinite.
</s>
</p>
<pb file="add_6782_f371v" o="371v" n="743"/>
<pb file="add_6782_f372" o="372" n="744"/>
<head xml:id="echoid-head140" xml:space="preserve" xml:lang="lat">
De Infinitis. Ratio Clava Herculis.
<lb/>[<emph style="it">tr:
On infinity. The ratio of the key of Hercules
</emph>]<lb/>
</head>
<pb file="add_6782_f372v" o="372v" n="745"/>
<pb file="add_6782_f373" o="373" n="746"/>
<head xml:id="echoid-head141" xml:space="preserve" xml:lang="lat">
De Infinitis.
<lb/>[<emph style="it">tr:
On infinity
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s1012" xml:space="preserve">
Suppose the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> doth touch the <lb/>
the circle in the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. &amp; touching <lb/>
in that point <emph style="super">in</emph> it only &amp; in no other <lb/>
point <emph style="st">it</emph>
<emph style="super">it</emph> toucheth, as Euclide suffi-<lb/>
ciently demonstrateth. Now I say <lb/>
there is (a point <foreign xml:lang="lat">deinceps</foreign>) a next <lb/>
poynt that doth not touch the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>.
</s>
</p>
<pb file="add_6782_f373v" o="373v" n="747"/>
<pb file="add_6782_f374" o="374" n="748"/>
<head xml:id="echoid-head142" xml:space="preserve" xml:lang="lat">
De Infinitis.
<lb/>[<emph style="it">tr:
On infinity
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s1013" xml:space="preserve">
Minimum. That will kill men by <lb/>
piercing &amp; running through. <lb/>
</s>
<s xml:id="echoid-s1014" xml:space="preserve">
Maximum. That which will presse men <lb/>
to death.
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1015" xml:space="preserve">
Unitas. Numeris unitatum. <lb/>
finitis <lb/>
infinitis
<lb/>[<emph style="it">tr:
Unity. The number of a unit. Finite. Infinite.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1016" xml:space="preserve">
Finites finitorum. <lb/>
Infinites finitorum. <lb/>
finites Infinitorum. <lb/>
Infinites Infinitorum. <lb/>
Infiniti infinitorum infinitum. <lb/>
Infiniti infinitorum finitum.
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1017" xml:space="preserve">
finitorum minimum. <lb/>
Infinitorum minimum. <lb/>
finitus minimorum. <lb/>
Infinites minimorum. <lb/>
finites finiti minimorum. <lb/>
Infinites finiti minimorum. <lb/>
Infinites finiti maximorum. <lb/>
Infinites infiniti maximorum. <lb/>
finiti. <lb/>
finitorum maximum .1. Infinitorum <lb/>
Infinitorum maximum. <lb/>
Infiniti.
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1018" xml:space="preserve">
Ratio Achilles
<lb/>[<emph style="it">tr:
The ratio of Achilles
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s1019" xml:space="preserve">
All the mistery of infinites lieth <lb/>
in <foreign xml:lang="lat">formati ratione
<emph style="super">unius</emph> unitatis</foreign> <lb/>
which is only respective, &amp; from <lb/>
where the knowledge &amp; import <lb/>
of <foreign xml:lang="lat">formalis ratio</foreign> of quantity <lb/>
doth spring.
</s>
</p>
<p>
<s xml:id="echoid-s1020" xml:space="preserve">
A finite space may be moved in infinite time.
</s>
<lb/>
<s xml:id="echoid-s1021" xml:space="preserve">
There is a [¿]conditioned[?]
motion that a finite space <emph style="super">given</emph> <lb/>
cannot be moved <emph style="super">in a finite time</emph> but in an infinite time.
</s>
<lb/>
<s xml:id="echoid-s1022" xml:space="preserve">
Also: that a finite space given cannot be moved
<emph style="super">in a finite time nor</emph> <lb/>
in an infinite time.
</s>
<lb/>
<s xml:id="echoid-s1023" xml:space="preserve">
Also: that an infinite space may be moved <lb/>
in a finite time.
</s>
<lb/>
<s xml:id="echoid-s1024" xml:space="preserve">
Also: that an infinite space <emph style="super">given</emph> may be moved not in <lb/>
a finite time but in an infinite time.
</s>
<lb/>
<s xml:id="echoid-s1025" xml:space="preserve">
Also: that an infinite space given, may not be <lb/>
moved either in an infinite time nor finite.
</s>
</p>
<p>
<s xml:id="echoid-s1026" xml:space="preserve">
Of contradictions that spring from diverse suppositions <lb/>
it cannot truly <emph style="st">[???]</emph>
<emph style="super">be</emph> sayd that the one parte
<emph style="st">doth [???]</emph> or <lb/>
other is false, for they are true consequently from <lb/>
there suppositions &amp; in that respect are both true. but <lb/>
that which followeth is, that one of the suppositions <lb/>
is necessarily false, from where one of the <lb/>
partes of the contradiction was inferred.
</s>
</p>
<p>
<s xml:id="echoid-s1027" xml:space="preserve">
As in the reason Achilles &amp; other <lb/>
reasons of Zeno &amp;c.
</s>
</p>
<pb file="add_6782_f374v" o="374v" n="749"/>
<pb file="add_6782_f375" o="375" n="750"/>
<pb file="add_6782_f375v" o="375v" n="751"/>
<pb file="add_6782_f376" o="376" n="752"/>
<pb file="add_6782_f376v" o="376v" n="753"/>
<pb file="add_6782_f377" o="377" n="754"/>
<pb file="add_6782_f377v" o="377v" n="755"/>
<pb file="add_6782_f378" o="378" n="756"/>
<pb file="add_6782_f378v" o="378v" n="757"/>
<pb file="add_6782_f379" o="379" n="758"/>
<pb file="add_6782_f379v" o="379v" n="759"/>
<pb file="add_6782_f380" o="380" n="760"/>
<head xml:id="echoid-head143" xml:space="preserve" xml:lang="lat">
Proponatur, per Artem Analyticam solvere et inde <lb/>
componere, hoc problema:
<lb/>[<emph style="it">tr:
It is proposed, by the analytic are to solve and thence compose this problem.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1028" xml:space="preserve">
<emph style="ul">Datam rectam terminatum: extrema ac media ratione secare.</emph>
<lb/>[<emph style="it">tr:
Given a finite straight line, to cut it in extreme and mean ratio.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1029" xml:space="preserve">
Sit data recta terminata <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. et <lb/>
ponatur secari in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> ita ut <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math> sit minor pars, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> maior.
<lb/>[<emph style="it">tr:
Let the given finite straight line be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> and let it be cut in the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>
so that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math> is the lesser part and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> the greater.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1030" xml:space="preserve">
Quoniam utraque pars ignota est, duæ possunt esse zeteses; etsi una <lb/>
sufficiat ad solutionem problematis.
<lb/>[<emph style="it">tr:
Since both parts are unkown, there are two possible zeteses, but one suffices for the solution of the problem.
</emph>]<lb/>
</s>
</p>
<head xml:id="echoid-head144" xml:space="preserve" xml:lang="lat">
Zetesis. 1.
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1031" xml:space="preserve">
ponatur primo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math> esse notum.
<lb/>[<emph style="it">tr:
first suppose <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math> is known.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1032" xml:space="preserve">
Tum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>a</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1033" xml:space="preserve">
Tres igitur proportionales erunt: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore three proportionals are: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1034" xml:space="preserve">
Inde resoluta analogia <lb/>
æaquatio erit.
<lb/>[<emph style="it">tr:
Whence having resolved the proportion, the equation will be:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1035" xml:space="preserve">
Et: per Antithesin:
<lb/>[<emph style="it">tr:
And by antithesis:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1036" xml:space="preserve">
Inde: Analogia.
<lb/>[<emph style="it">tr:
Whence, the ratio.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1037" xml:space="preserve">
Ubi datur media proportionalis et adgregatum exremarum, <emph style="st">per</emph> <lb/>
ad Exegesin:
<lb/>[<emph style="it">tr:
Where there is given the mean proportional and the sum of the extremes, the resolution:
</emph>]<lb/>
</s>
</p>
<head xml:id="echoid-head145" xml:space="preserve" xml:lang="lat">
Zetesis. 2.
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1038" xml:space="preserve">
ponatur secundo, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> esse notam.
<lb/>[<emph style="it">tr:
suppose the second, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>, is known.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1039" xml:space="preserve">
Tum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math> erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>-</mo><mi>a</mi><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math> will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>-</mo><mi>a</mi><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1040" xml:space="preserve">
Tres igitur proportionales erunt. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>-</mo><mi>a</mi><mi>c</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>b</mi><mi>c</mi></mstyle></math>
<lb/>[<emph style="it">tr:
Therefore three proportionals will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>-</mo><mi>a</mi><mi>c</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>b</mi><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1041" xml:space="preserve">
Inde: resoluta analogia: erit.
<lb/>[<emph style="it">tr:
Whence the resolution of the ratio will be:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1042" xml:space="preserve">
Et: per Antithesin.
<lb/>[<emph style="it">tr:
And by antihesis:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1043" xml:space="preserve">
Inde: analogia: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</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>b</mi><mi>c</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Whence the ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</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>b</mi><mi>c</mi></mstyle></math> : <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1044" xml:space="preserve">
Ubi datur media proportionalis et differentia extremarum, <lb/>
ad Exegesin.
<lb/>[<emph style="it">tr:
Where there is given a mean proportional and the difference of the extremes, the resolution.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f380v" o="380v" n="761"/>
<pb file="add_6782_f381" o="381" n="762"/>
<pb file="add_6782_f381v" o="381v" n="763"/>
<pb file="add_6782_f382" o="382" n="764"/>
<pb file="add_6782_f382v" o="382v" n="765"/>
<pb file="add_6782_f383" o="383" n="766"/>
<p>
<s xml:id="echoid-s1045" xml:space="preserve">
to devide which by <lb/>
[???] a mean <lb/>
proportional. <lb/>
[???] <lb/>
the first being given is <lb/>
the same [???] all 3 to [???] <lb/>
the proportionals.
</s>
</p>
<pb file="add_6782_f383v" o="383v" n="767"/>
<pb file="add_6782_f384" o="384" n="768"/>
<pb file="add_6782_f384v" o="384v" n="769"/>
<pb file="add_6782_f385" o="385" n="770"/>
<pb file="add_6782_f385v" o="385v" n="771"/>
<pb file="add_6782_f386" o="386" n="772"/>
<pb file="add_6782_f386v" o="386v" n="773"/>
<pb file="add_6782_f387" o="387" n="774"/>
<pb file="add_6782_f387v" o="387v" n="775"/>
<pb file="add_6782_f388" o="388" n="776"/>
<div xml:id="echoid-div236" type="page_commentary" level="2" n="236">
<p>
<s xml:id="echoid-s1046" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1046" xml:space="preserve">
For a general explanation of the method see Add MS 6782, f. 399. <lb/>
The example on this page is a quartic equation with a fourth power, a linear term and a square,
the equation from Problem 6 of Viète's
<emph style="it">De numerosa potestatum resolutione</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head146" xml:space="preserve" xml:lang="lat">
12.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s1048" xml:space="preserve">
prob. 6. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>d</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>f</mi><mi>f</mi><mi>f</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Problem 6. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>d</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>f</mi><mi>f</mi><mi>f</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1049" xml:space="preserve">
Unicum <lb/>
Vietæ exemplum.
<lb/>[<emph style="it">tr:
Viète's only example.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1050" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>2</mn><mn>0</mn><mn>0</mn><mo>,</mo><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>0</mn><mn>0</mn><mi>a</mi><mo>=</mo><mn>4</mn><mn>4</mn><mn>9</mn><mn>3</mn><mn>7</mn><mn>6</mn></mstyle></math>
</s>
<lb/>
<s xml:id="echoid-s1051" xml:space="preserve">
Species <lb/>
canonica.
<lb/>[<emph style="it">tr:
Canonical form.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1052" xml:space="preserve">
Resolutio.
<lb/>[<emph style="it">tr:
Solution.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f388v" o="388v" n="777"/>
<pb file="add_6782_f389" o="389" n="778"/>
<div xml:id="echoid-div237" type="page_commentary" level="2" n="237">
<p>
<s xml:id="echoid-s1053" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1053" xml:space="preserve">
For a general explanation of the method see Add MS 6782, f. 399. <lb/>
The example on this page is a quartic equation with only a fourth power and a cube term,
the equation from Problem 5 of Viète's
<emph style="it">De numerosa potestatum resolutione</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head147" xml:space="preserve" xml:lang="lat">
11.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s1055" xml:space="preserve">
prob. 5. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Problem 5. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1056" xml:space="preserve">
Unicum <lb/>
Vietæ exemplum.
<lb/>[<emph style="it">tr:
Viète's only exmaple.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1057" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>0</mn><mo>,</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>0</mn><mn>0</mn><mi>a</mi><mo>=</mo><mn>4</mn><mn>7</mn><mn>0</mn><mn>0</mn><mn>1</mn><mn>6</mn></mstyle></math>
</s>
<lb/>
<s xml:id="echoid-s1058" xml:space="preserve">
Species <lb/>
canonica.
<lb/>[<emph style="it">tr:
Canonical form.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1059" xml:space="preserve">
Resolutio.
<lb/>[<emph style="it">tr:
solution.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1060" xml:space="preserve">
Alterum exemplum nostrum: <lb/>
quod per divisionem.
<lb/>[<emph style="it">tr:
Another example of my own, done by division.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1061" xml:space="preserve">
Resolutio.
<lb/>[<emph style="it">tr:
Solution.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1062" xml:space="preserve">
pro 1<emph style="super">a</emph> figura <lb/>
Divide 44 <lb/>
per. 3 <lb/>
quotiens. 14. <lb/>
cuius maximus cubus 8 <lb/>
latus 2. <lb/>
pro 1<emph style="super">a</emph> figura <lb/>
si caetera <lb/>
consentiunt
<lb/>[<emph style="it">tr:
For the first figure, divide 44 by 3; the quotient is 14, whose greatest cube is 8, with side 2,
[to be taken] for the first figure, if the rest agree.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f389v" o="389v" n="779"/>
<pb file="add_6782_f390" o="390" n="780"/>
<div xml:id="echoid-div238" type="page_commentary" level="2" n="238">
<p>
<s xml:id="echoid-s1063" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1063" xml:space="preserve">
For a general explanation of the method see Add MS 6782, f. 399. <lb/>
The example on this page is a quartic equation with only a fourth power and a linear term,
the second equation from Problem 4 of Viète's
<emph style="it">De numerosa potestatum resolutione</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head148" xml:space="preserve" xml:lang="lat">
10.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s1065" xml:space="preserve">
prob. 4. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Problem 4. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1066" xml:space="preserve">
2. <lb/>
Vietæ exemplum. <lb/>
quod Per divisionem.
<lb/>[<emph style="it">tr:
Viète's second example, done by division.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1067" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>0</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>7</mn><mn>3</mn><mn>1</mn><mo>,</mo><mn>7</mn><mn>7</mn><mn>6</mn></mstyle></math>
</s>
<lb/>
<s xml:id="echoid-s1068" xml:space="preserve">
Species <lb/>
canonica.
<lb/>[<emph style="it">tr:
Canonical form.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1069" xml:space="preserve">
Resolutio.
<lb/>[<emph style="it">tr:
Solution.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f390v" o="390v" n="781"/>
<pb file="add_6782_f391" o="391" n="782"/>
<div xml:id="echoid-div239" type="page_commentary" level="2" n="239">
<p>
<s xml:id="echoid-s1070" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1070" xml:space="preserve">
For a general explanation of the method see Add MS 6782, f. 399. <lb/>
The example on this page is a quartic equation with only a fourth power and a linear term,
the first equation from Problem 4 of Viète's
<emph style="it">De numerosa potestatum resolutione</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head149" xml:space="preserve" xml:lang="lat">
9.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s1072" xml:space="preserve">
prob. 4. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Problem 4. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi><mi>z</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1073" xml:space="preserve">
1. <lb/>
Vietæ exemplum.
<lb/>[<emph style="it">tr:
Viète's first example.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1074" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>3</mn><mn>5</mn><mn>5</mn><mo>,</mo><mn>7</mn><mn>7</mn><mn>6</mn></mstyle></math>
</s>
<lb/>
<s xml:id="echoid-s1075" xml:space="preserve">
Species <lb/>
canonica.
<lb/>[<emph style="it">tr:
Canonical form.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1076" xml:space="preserve">
Resolutio.
<lb/>[<emph style="it">tr:
Solution.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f391v" o="391v" n="783"/>
<pb file="add_6782_f392" o="392" n="784"/>
<div xml:id="echoid-div240" type="page_commentary" level="2" n="240">
<p>
<s xml:id="echoid-s1077" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1077" xml:space="preserve">
For a general explanation of the method see Add MS 6782, f. 399. <lb/>
The example on this page is a cubic equation without a linear term,
the second equation from Problem 3 of Viète's
<emph style="it">De numerosa potestatum resolutione</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head150" xml:space="preserve" xml:lang="lat">
8.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s1079" xml:space="preserve">
prob. 3. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Problem 3. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1080" xml:space="preserve">
2. <lb/>
Vietæ exemplum <lb/>
quod per divisionem.
<lb/>[<emph style="it">tr:
Viète's second example, done by division.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1081" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>,</mo><mi>a</mi><mi>a</mi><mo>=</mo><mn>5</mn><mo>,</mo><mn>7</mn><mn>7</mn><mn>3</mn><mo>,</mo><mn>8</mn><mn>2</mn><mn>4</mn></mstyle></math>
</s>
<lb/>
<s xml:id="echoid-s1082" xml:space="preserve">
Species <lb/>
canonica.
<lb/>[<emph style="it">tr:
Canonical form.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1083" xml:space="preserve">
Resolutio.
<lb/>[<emph style="it">tr:
Solution.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1084" xml:space="preserve">
Nota.
<lb/>[<emph style="it">tr:
Note.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1085" xml:space="preserve">
prima figura acquiritur <lb/>
per divisionem ac si <lb/>
Species canonica <lb/>
esset: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>d</mi><mo>+</mo><mi>b</mi><mo maxsize="1">)</mo><mo>×</mo><mi>b</mi><mi>b</mi></mstyle></math> quæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>b</mi><mi>b</mi><mi>d</mi><mo>+</mo><mi>b</mi><mi>b</mi><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
the first figure is acquired by division as if the canonical form were <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>d</mi><mo>+</mo><mi>b</mi><mo maxsize="1">)</mo><mo>×</mo><mi>b</mi><mi>b</mi></mstyle></math> quæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>b</mi><mi>b</mi><mi>d</mi><mo>+</mo><mi>b</mi><mi>b</mi><mi>b</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1086" xml:space="preserve">
hoc est si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>+</mo><mi>b</mi></mstyle></math> sit <lb/>
divisor <lb/>
Quotiens erit, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math>
<lb/>[<emph style="it">tr:
that is, if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>+</mo><mi>b</mi></mstyle></math> is the divisor, the quotient will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1087" xml:space="preserve">
Et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> latus erit 1<emph style="super">a</emph> figura
<lb/>[<emph style="it">tr:
And the square-root <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> will be the first figure.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1088" xml:space="preserve">
Ut hic Quotiens est 5. <lb/>
cuius latus <emph style="super">quadratum</emph> 2. pro <lb/>
prima figura.
<lb/>[<emph style="it">tr:
As here the quotient is 5, whose square-root is 2 for the first figure.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1089" xml:space="preserve">
Vide Vietam.
<lb/>[<emph style="it">tr:
See Viète.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f392v" o="392v" n="785"/>
<pb file="add_6782_f393" o="393" n="786"/>
<div xml:id="echoid-div241" type="page_commentary" level="2" n="241">
<p>
<s xml:id="echoid-s1090" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1090" xml:space="preserve">
For a general explanation of the method see Add MS 6782, f. 399. <lb/>
The example on this page is a cubic equation without a linear term,
the first equation from Problem 3 of Viète's
<emph style="it">De numerosa potestatum resolutione</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head151" xml:space="preserve" xml:lang="lat">
7.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s1092" xml:space="preserve">
prob. 3. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Problem 3. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1093" xml:space="preserve">
1. <lb/>
Vietæ exemplum
<lb/>[<emph style="it">tr:
Viète's first example
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1094" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>3</mn><mn>0</mn><mo>,</mo><mi>a</mi><mi>a</mi><mo>=</mo><mn>8</mn><mn>6</mn><mo>,</mo><mn>2</mn><mn>2</mn><mn>0</mn><mo>,</mo><mn>2</mn><mn>8</mn><mn>8</mn></mstyle></math>.
</s>
<lb/>
<s xml:id="echoid-s1095" xml:space="preserve">
Species <lb/>
canonica.
<lb/>[<emph style="it">tr:
Canonical form.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1096" xml:space="preserve">
Resolutio.
<lb/>[<emph style="it">tr:
Solution.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1097" xml:space="preserve">
Jam 43. fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Now <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> becomes 43.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f393v" o="393v" n="787"/>
<pb file="add_6782_f394" o="394" n="788"/>
<div xml:id="echoid-div242" type="page_commentary" level="2" n="242">
<p>
<s xml:id="echoid-s1098" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1098" xml:space="preserve">
For a general explanation of the method see Add MS 6782, f. 399. <lb/>
The example on this page is a cubic equation without a square term,
the second equation from Problem 2 of Viète's
<emph style="it">De numerosa potestatum resolutione</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head152" xml:space="preserve" xml:lang="lat">
6.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s1100" xml:space="preserve">
prob. 2. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Problem 2. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>.
</emph>]<lb/></s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1101" xml:space="preserve">
2. <lb/>
Vietæ exemplum
<lb/>[<emph style="it">tr:
Viète's second example.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1102" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>9</mn><mn>5</mn><mn>4</mn><mn>0</mn><mn>0</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>8</mn><mn>1</mn><mn>9</mn><mo>,</mo><mn>4</mn><mn>5</mn><mn>9</mn></mstyle></math>
</s>
<lb/>
<s xml:id="echoid-s1103" xml:space="preserve">
Species <lb/>
canonica
<lb/>[<emph style="it">tr:
Canonical form
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1104" xml:space="preserve">
Resolutio:
<lb/>[<emph style="it">tr:
Solution.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1105" xml:space="preserve">
per divisionem.
<lb/>[<emph style="it">tr:
By division.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f394v" o="394v" n="789"/>
<pb file="add_6782_f395" o="395" n="790"/>
<div xml:id="echoid-div243" type="page_commentary" level="2" n="243">
<p>
<s xml:id="echoid-s1106" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1106" xml:space="preserve">
For a general explanation of the method see Add MS 6782, f. 399. <lb/>
The example on this page is a cubic equation without a square term,
the first equation from Problem 2 of Viète's
<emph style="it">De numerosa potestatum resolutione</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head153" xml:space="preserve" xml:lang="lat">
5.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s1108" xml:space="preserve">
prob. 2. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Problem 2. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1109" xml:space="preserve">
1. <lb/>
Vietæ exemplum
<lb/>[<emph style="it">tr:
Viète's first example.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1110" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>3</mn><mn>0</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>1</mn><mn>4</mn><mn>3</mn><mn>5</mn><mn>6</mn><mn>1</mn><mn>9</mn><mn>7</mn></mstyle></math>.
</s>
<lb/>
<s xml:id="echoid-s1111" xml:space="preserve">
Species <lb/>
canonica.
<lb/>[<emph style="it">tr:
Canonical form.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1112" xml:space="preserve">
Vel:
<lb/>[<emph style="it">tr:
Or:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1113" xml:space="preserve">
Resolutio.
<lb/>[<emph style="it">tr:
Solution.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1114" xml:space="preserve">
Vide supra. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
See above at A.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1115" xml:space="preserve">
Jam 24 fiat. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Now <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> becomes 24.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f395v" o="395v" n="791"/>
<pb file="add_6782_f396" o="396" n="792"/>
<div xml:id="echoid-div244" type="page_commentary" level="2" n="244">
<p>
<s xml:id="echoid-s1116" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1116" xml:space="preserve">
For a general explanation of the method see Add MS 6782, f. 399. <lb/>
Here the method is applied to a cubic equation without a square term.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head154" xml:space="preserve" xml:lang="lat">
4.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s1118" xml:space="preserve">
prob. 2. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Problem 2. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1119" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>3</mn><mn>5</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>2</mn><mn>2</mn><mn>9</mn><mn>3</mn><mn>2</mn></mstyle></math>
</s>
<lb/>
<s xml:id="echoid-s1120" xml:space="preserve">
Species <lb/>
canonica
<lb/>[<emph style="it">tr:
Canonical form
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1121" xml:space="preserve">
Hoc est:
<lb/>[<emph style="it">tr:
That is:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1122" xml:space="preserve">
Vel.
<lb/>[<emph style="it">tr:
Or.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1123" xml:space="preserve">
Resolutio.
<lb/>[<emph style="it">tr:
Solution.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f396v" o="396v" n="793"/>
<pb file="add_6782_f397" o="397" n="794"/>
<div xml:id="echoid-div245" type="page_commentary" level="2" n="245">
<p>
<s xml:id="echoid-s1124" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1124" xml:space="preserve">
For a general explanation of the method see Add MS 6782, f. 399. <lb/>
In the example on this page, the large size of the coefficient in relation to the root means that
the term <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mn>6</mn><mn>2</mn><mi>b</mi></mstyle></math> must be taken into account in determining the first digit.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head155" xml:space="preserve" xml:lang="lat">
3.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1126" xml:space="preserve">
prob. 1. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>z</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Problem 1. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>z</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1127" xml:space="preserve">
Casus 2<emph style="super">a</emph>. <lb/>
per divisionem
<lb/>[<emph style="it">tr:
Case 2, by division
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1128" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mn>7</mn><mn>6</mn><mn>2</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>2</mn><mn>2</mn><mn>1</mn><mn>2</mn><mn>0</mn></mstyle></math>
</s>
<lb/>
<s xml:id="echoid-s1129" xml:space="preserve">
Species <lb/>
canonica <lb/>
ut supra.
<lb/>[<emph style="it">tr:
Canonical form as above.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1130" xml:space="preserve">
Vel:
<lb/>[<emph style="it">tr:
Or:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1131" xml:space="preserve">
Resolutio.
<lb/>[<emph style="it">tr:
Solution
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1132" xml:space="preserve">
Quoniam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mo>&gt;</mo><mn>2</mn></mstyle></math> <lb/>
fiat devolutio.
<lb/>[<emph style="it">tr:
Because <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mo>&gt;</mo><mn>2</mn></mstyle></math> it becomes a devolution.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1133" xml:space="preserve">
Vietæ exemplum. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mn>9</mn><mn>5</mn><mn>4</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>1</mn><mn>8</mn><mn>4</mn><mn>8</mn><mn>7</mn></mstyle></math>
<lb/>[<emph style="it">tr:
Viète's example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mn>9</mn><mn>5</mn><mn>4</mn><mi>a</mi><mo>=</mo><mn>1</mn><mn>8</mn><mn>4</mn><mn>8</mn><mn>7</mn></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f397v" o="397v" n="795"/>
<pb file="add_6782_f398" o="398" n="796"/>
<div xml:id="echoid-div246" type="page_commentary" level="2" n="246">
<p>
<s xml:id="echoid-s1134" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1134" xml:space="preserve">
For a general explanation of the method see Add MS 6782, f. 399. <lb/>
The equation on this page has a 3-digit solution. The first digit is found by inspection to be 2 (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mn>2</mn><mn>0</mn><mn>0</mn></mstyle></math>).
The second digit, is found to be 4 (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>4</mn><mn>0</mn></mstyle></math>).
The process is now repeated, treating <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> as a single quantity, to find the third digit, again labelled <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
This is found to be 3.
For Harriot's owon description of this process see Add MS 6784, f. 408.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head156" xml:space="preserve" xml:lang="lat">
2.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1136" xml:space="preserve">
prob. 1. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>z</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Problem 1. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>z</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1137" xml:space="preserve">
Exemplum primum <lb/>
Vietæ.
<lb/>[<emph style="it">tr:
Viète's first example
</emph>]<lb/>
[<emph style="it">Note:
This is Problem 1 from Viète's <emph style="it">De numerosa potestatum resolutione</emph>.
 </emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s1138" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mn>7</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>6</mn><mn>0</mn><mn>7</mn><mn>5</mn><mn>0</mn></mstyle></math>
</s>
<lb/>
<s xml:id="echoid-s1139" xml:space="preserve">
Species <lb/>
canonica. <lb/>
ut supra.
<lb/>[<emph style="it">tr:
Canonical form as above.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1140" xml:space="preserve">
hoc est:
<lb/>[<emph style="it">tr:
that is:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1141" xml:space="preserve">
Resolutio Vietana. paucis mutatis
<lb/>[<emph style="it">tr:
Viète's solution, a little changed
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f398v" o="398v" n="797"/>
<pb file="add_6782_f399" o="399" n="798"/>
<div xml:id="echoid-div247" type="page_commentary" level="2" n="247">
<p>
<s xml:id="echoid-s1142" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1142" xml:space="preserve">
This is the first of a set of 12 pages on extracting roots of positively affected equations,
that is, equations where all the powers following the first are positive.
Such equations have one, and only one, positive root. <lb/>
The work is closely based on Problems 1 to 6 in
Viète, <emph style="it">De numerosa potestatum ad exegesin resolutione</emph> (1600);
Harriot's heading 'De numerosa potestatum resolutione' directly echoes Viète's. <lb/>
The method works by finding each digit of the root in turn.
Suppose that a required root <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> of is of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>,
where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> represents multiples of 10 and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> represents units.
The first digit is found by inspection. The canonical form then shows how to estimate <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. <lb/>
In the problem on this page, for example, Harriot first takes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> to be 40, but quickly finds that this is too large.
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> must be 30. Subtracting the known values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math> (= 900) and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi></mstyle></math> (= 720) from 2356
leaves him with 736, which according to the canonical form must correspond to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>d</mi><mo>+</mo><mi>c</mi><mi>c</mi></mstyle></math>.
A first estimate for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is found by dividing 736 by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math> (= 84). The integer part of the quotient is 8.
In fact <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>8</mn></mstyle></math> satisfies the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>d</mi><mo>+</mo><mi>c</mi><mi>c</mi><mo>=</mo><mn>7</mn><mn>3</mn><mn>6</mn></mstyle></math> exactly and so the process is complete. <lb/>
Later examples become more complicated but follow the same basic procedure. <lb/>
For further discussion see See Stedall 2003, 45–62 and 292, and Stedall 2011, 29–31.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head157" xml:space="preserve" xml:lang="lat">
1.) De numerosa potestatum resolutione. Vieta. fol. 7. b.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers. Viète, folio 7b.
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s1144" xml:space="preserve">
prob. 1. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>z</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Problem 1. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>z</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1145" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mn>2</mn><mn>4</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>2</mn><mn>3</mn><mn>5</mn><mn>6</mn></mstyle></math>.
</s>
<lb/>
<s xml:id="echoid-s1146" xml:space="preserve">
Species <lb/>
canonica
<lb/>[<emph style="it">tr:
Canonical form
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1147" xml:space="preserve">
Hoc est:
<lb/>[<emph style="it">tr:
That is:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1148" xml:space="preserve">
vel:
<lb/>[<emph style="it">tr:
or:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1149" xml:space="preserve">
Genesis Vulgaris
<lb/>[<emph style="it">tr:
Common derivation
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1150" xml:space="preserve">
Genesis Specialis <lb/>
seu canonica
<lb/>[<emph style="it">tr:
Specific or canonical derivation.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1151" xml:space="preserve">
Resolutio secundum methodum Vietanam, <lb/>
paucis mutatis
<lb/>[<emph style="it">tr:
Solution according to the method of Viète, a little changed
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1152" xml:space="preserve">
non potest auferri. <lb/>
operatio igitur iteranda <lb/>
et fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. minor. videlicet. 3.
<lb/>[<emph style="it">tr:
cannot be subtracted; the work must therefore be repeated making <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> smaller, namely, 3.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1153" xml:space="preserve">
Aliter.
<lb/>[<emph style="it">tr:
Another way.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f399v" o="399v" n="799"/>
<pb file="add_6782_f400" o="400" n="800"/>
<div xml:id="echoid-div248" type="page_commentary" level="2" n="248">
<p>
<s xml:id="echoid-s1154" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1154" xml:space="preserve">
Following from the general treatment in Add MS 6782, f. 403, f. 402, f. 401,
of avulsed quartics with no square or linear term,
Harriot here solves the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>4</mn><mn>8</mn><mn>1</mn><mn>5</mn><mn>4</mn><mn>4</mn><mo>=</mo><mn>6</mn><mn>5</mn><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> for both roots (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn><mn>8</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>5</mn><mn>7</mn></mstyle></math>).
He also shows how either root may be obtained from the other. <lb/>
The equation is taken from Problem 20 of Viète,
<emph style="it">De numerosa potestatum ad exegesin resolutione</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head158" xml:space="preserve" xml:lang="lat">
c.17.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1156" xml:space="preserve">
prob. 20. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex.
<lb/>[<emph style="it">tr:
Problem 20. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1157" xml:space="preserve">
canon ad <lb/>
resolutionem.
<lb/>[<emph style="it">tr:
Canonical form for the solution.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1158" xml:space="preserve">
Resolutio.
<lb/>[<emph style="it">tr:
Solution.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1159" xml:space="preserve">
Eductio radicis <lb/>
minoris.
<lb/>[<emph style="it">tr:
Extraction of the smaller root.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1160" xml:space="preserve">
Radix igitur <lb/>
minor. 38.
<lb/>[<emph style="it">tr:
Therefore the smaller root is 38.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1161" xml:space="preserve">
Quæratur maior.
<lb/>[<emph style="it">tr:
The larger root is sought.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1162" xml:space="preserve">
Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, minor <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> maior.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the smaller root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> the larger.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1163" xml:space="preserve">
Vieta aliter <lb/>
ut pag: supra.
<lb/>[<emph style="it">tr:
Viète otherwise, as in the page above.
</emph>]<lb/>
[<emph style="it">Note:
The page above is Add MS 6782, f. 402.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1164" xml:space="preserve">
Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. 57.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>5</mn><mn>7</mn></mstyle></math> is given.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1165" xml:space="preserve">
Eductio radicis <lb/>
maioris.
<lb/>[<emph style="it">tr:
Extraction of the larger root.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1166" xml:space="preserve">
Radix igitur maior. 57.
<lb/>[<emph style="it">tr:
Therefore the larger root is 57.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1167" xml:space="preserve">
Quæratur iam minor.
<lb/>[<emph style="it">tr:
Now the smaller root is sought.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1168" xml:space="preserve">
Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> minor, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> minor.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> be the smaller root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> the larger.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1169" xml:space="preserve">
Vieta aliter <lb/>
ut pag: supra.
<lb/>[<emph style="it">tr:
Viète otherwise as in the page above.
</emph>]<lb/>
[<emph style="it">Note:
The page above is Add MS 6782, f. 402.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1170" xml:space="preserve">
Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. 38.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mn>3</mn><mn>8</mn></mstyle></math> is given.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f400v" o="400v" n="801"/>
<pb file="add_6782_f401" o="401" n="802"/>
<div xml:id="echoid-div249" type="page_commentary" level="2" n="249">
<p>
<s xml:id="echoid-s1171" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1171" xml:space="preserve">
In this final page of Section c, Harriot argues that the first figure of the larger root must be 5.
Assuming that it is either 4 or 6 will lead to a contradiction.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head159" xml:space="preserve" xml:lang="lat">
c.18.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1173" xml:space="preserve">
prob. 20. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>,</mo><mn>4</mn><mn>8</mn><mn>1</mn><mo>,</mo><mn>5</mn><mn>4</mn><mn>4</mn><mo>=</mo><mn>6</mn><mn>5</mn><mo>,</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>.. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex.
<lb/>[<emph style="it">tr:
Problem 20. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>4</mn><mn>8</mn><mn>1</mn><mn>5</mn><mn>4</mn><mn>4</mn><mo>=</mo><mn>6</mn><mn>5</mn><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>.. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1174" xml:space="preserve">
<emph style="st">Additementum <lb/>
nostrum</emph> <lb/>
Absurdum consequens in eductione lateris maioris <lb/>
si prima figura sit minor vel maior. 5.
<lb/>[<emph style="it">tr:
Nonsensical consequences in the extraction of the larger root if the first figure is less than or greater than 5.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1175" xml:space="preserve">
B. Divisor <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mn>4</mn><mn>1</mn><mn>0</mn></mstyle></math>
<lb/>[<emph style="it">tr:
The divisor is 5410.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1176" xml:space="preserve">
Absurdum.
<lb/>[<emph style="it">tr:
Contradiction
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1177" xml:space="preserve">
Nam cum Residuum supra <lb/>
habent signum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo></mstyle></math> <lb/>
et Divisor <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo></mstyle></math> <lb/>
Parabola erit etiam, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo></mstyle></math>.
<lb/>[<emph style="it">tr:
For with the above residue we have a <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo></mstyle></math> sign, while the divisor is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo></mstyle></math>,
so the comparison is also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1178" xml:space="preserve">
Prima figura igitur <lb/>
non est 4.
<lb/>[<emph style="it">tr:
Therefore the first figure is not 4.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1179" xml:space="preserve">
B. Divisor <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mn>1</mn><mn>7</mn><mn>2</mn><mn>0</mn><mn>7</mn><mn>5</mn></mstyle></math>
<lb/>[<emph style="it">tr:
The divisor is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mn>1</mn><mn>7</mn><mn>2</mn><mn>0</mn><mn>7</mn><mn>5</mn></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1180" xml:space="preserve">
Absurdum.
<lb/>[<emph style="it">tr:
Contradiction.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1181" xml:space="preserve">
Nam cum residuum supra <lb/>
habent signum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo></mstyle></math> <lb/>
et divisor, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo></mstyle></math> <lb/>
parabola erit etiam, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo></mstyle></math>.
<lb/>[<emph style="it">tr:
For while the above residue has the sign <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo></mstyle></math> and the divisor <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo></mstyle></math>, the comparison is also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1182" xml:space="preserve">
Prima figura igitur <lb/>
non est, 6.
<lb/>[<emph style="it">tr:
Therefore the first figure is not 6.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1183" xml:space="preserve">
Erit igitur 5. per limitum <lb/>
præfinitiones, sicut <lb/>
per exemplum est explicatum.
<lb/>[<emph style="it">tr:
Therefore it will be 5, by the determinations of limits, as explained by the example.
</emph>]<lb/>
</s>
</p>
<head xml:id="echoid-head160" xml:space="preserve" xml:lang="lat">
Nota.
<lb/>[<emph style="it">tr:
Note
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1184" xml:space="preserve">
Operatio pro eductione lateris maioris aliquando secundum partem <lb/>
est similis operationem pro eductione lateris minoris.
<lb/>[<emph style="it">tr:
In the operation for extracting the larger root,
sometimes the second part is similar to the operation for extracting the smaller root.
</emph>]<lb/>
</s>
<s xml:id="echoid-s1185" xml:space="preserve">
Nimirum si <lb/>
utraque latera consentiant in primus figurus
<emph style="super">et numero figurum</emph>.
Evidently, if both roots agree in the first figure and the number of figures.
</s>
<s xml:id="echoid-s1186" xml:space="preserve">
Ut si latus minus <lb/>
sit 23. maius 24.
<lb/>[<emph style="it">tr:
As if the smaller root is 23, the larger is 24.
</emph>]<lb/>
</s>
<s xml:id="echoid-s1187" xml:space="preserve">
Ita si minus 343, maius 347. est sic de cæteris.
<lb/>[<emph style="it">tr:
Thus if the smaller root is 343, the larger 347, and so on for others.
</emph>]<lb/>
</s>
<s xml:id="echoid-s1188" xml:space="preserve">
Sed si dissentiant in primum figura: dissimiles erunt operationes totaliter <lb/>
ut in exemplis antecedentis, in his chartis expositis.
<lb/>[<emph style="it">tr:
But if they do not agree in the first figure, the operations will be completely different,
as in the preceding examples explained in these sheets.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f401v" o="401v" n="803"/>
<pb file="add_6782_f402" o="402" n="804"/>
<div xml:id="echoid-div250" type="page_commentary" level="2" n="250">
<p>
<s xml:id="echoid-s1189" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1189" xml:space="preserve">
On this page Harriot compares his own method with that of Viète, in Problem 20 of
<emph style="it">De numerosa potestatum ad exegesin resolutione</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head161" xml:space="preserve" xml:lang="lat">
c.16.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1191" xml:space="preserve">
prob. 20. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex.
<lb/>[<emph style="it">tr:
Problem 20. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1192" xml:space="preserve">
Species canonica <lb/>
ad radices inæquales.
<lb/>[<emph style="it">tr:
Canonical form for unequal roots.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1193" xml:space="preserve">
Aliter quam supra, <lb/>
et ut Vieta.
<lb/>[<emph style="it">tr:
Another way from that above, and as Viète.
</emph>]<lb/>
[<emph style="it">Note:
By 'above' Harriot means the working given in Add MS 6782, f. 404, f. 403.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1194" xml:space="preserve">
Si una radix sit nota, <lb/>
altera erit cognita.
<lb/>[<emph style="it">tr:
If one roots is known, the other will be known.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1195" xml:space="preserve">
4<emph style="super">or</emph> continue proportionales.
Four continued proportionals.
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1196" xml:space="preserve">
Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> nota. Quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is sought.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1197" xml:space="preserve">
Sint continue proportionales
<lb/>[<emph style="it">tr:
Let there be continued proportinals
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1198" xml:space="preserve">
datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> et inde <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> is given and thence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1199" xml:space="preserve">
erit: <lb/>
inde:
<lb/>[<emph style="it">tr:
We will have: <lb/>
thence:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1200" xml:space="preserve">
Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> nota. Quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is sought.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1201" xml:space="preserve">
Sint continue proportionales
<lb/>[<emph style="it">tr:
Let there be continued proportinals
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1202" xml:space="preserve">
datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> et inde, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> is given, and thence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1203" xml:space="preserve">
erit: <lb/>
inde:
<lb/>[<emph style="it">tr:
We will have: <lb/>
thence:
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f402v" o="402v" n="805"/>
<pb file="add_6782_f403" o="403" n="806"/>
<div xml:id="echoid-div251" type="page_commentary" level="2" n="251">
<p>
<s xml:id="echoid-s1204" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1204" xml:space="preserve">
On this page, Harriot continues his general treatment of equations of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>,
begun on the previous page. <lb/>
The numerical example at the end, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>4</mn><mn>8</mn><mn>1</mn><mn>5</mn><mn>4</mn><mn>4</mn><mo>=</mo><mn>6</mn><mn>5</mn><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, is from Problem 20 of Viète,
<emph style="it">De numerosa potestatum ad exegesin resolutione</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head162" xml:space="preserve" xml:lang="lat">
c.15.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1206" xml:space="preserve">
prob. 20. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex.
<lb/>[<emph style="it">tr:
Problem 20. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1207" xml:space="preserve">
Species canonica <lb/>
ad radices inæquales.
<lb/>[<emph style="it">tr:
Canonical form for unequal roots.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1208" xml:space="preserve">
Si una radix sit nota, <lb/>
altera erit cognita.
<lb/>[<emph style="it">tr:
If one root is known, the other will be known.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1209" xml:space="preserve">
sit, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> nota. Quæaeritur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is sought.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1210" xml:space="preserve">
Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1211" xml:space="preserve">
Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1212" xml:space="preserve">
sit, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> nota. Quæaeritur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is sought.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1213" xml:space="preserve">
Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1214" xml:space="preserve">
Unde eadem æquatio:
<lb/>[<emph style="it">tr:
Whence the same equation:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1215" xml:space="preserve">
Datur <emph style="st">igitur</emph> igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1216" xml:space="preserve">
Poristicum
<lb/>[<emph style="it">tr:
Proof
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1217" xml:space="preserve">
Quod,
<lb/>[<emph style="it">tr:
Because
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1218" xml:space="preserve">
Est enim: est igitur.
<lb/>[<emph style="it">tr:
Indeed it is. Therefore it is so.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1219" xml:space="preserve">
Pro exemplo ad resolutionem.
<lb/>[<emph style="it">tr:
According to this example for the solution.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1220" xml:space="preserve">
In numeris, sit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>.</mo><mn>3</mn><mn>8</mn></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>.</mo><mn>5</mn><mn>7</mn><mo>.</mo></mstyle></math>
<lb/>[<emph style="it">tr:
In numbers let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mn>3</mn><mn>8</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>5</mn><mn>7</mn></mstyle></math>.
</emph>]<lb/>
</s>
<s xml:id="echoid-s1221" xml:space="preserve">
Hoc est: <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>,</mo><mn>4</mn><mn>8</mn><mn>1</mn><mo>,</mo><mn>5</mn><mn>4</mn><mn>4</mn><mo>=</mo><mn>6</mn><mn>5</mn><mo>,</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
That is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>4</mn><mn>8</mn><mn>1</mn><mn>5</mn><mn>4</mn><mn>4</mn><mo>=</mo><mn>6</mn><mn>5</mn><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1222" xml:space="preserve">
Limites radicum.
<lb/>[<emph style="it">tr:
The limits of the roots.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f403v" o="403v" n="807"/>
<pb file="add_6782_f404" o="404" n="808"/>
<div xml:id="echoid-div252" type="page_commentary" level="2" n="252">
<p>
<s xml:id="echoid-s1223" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1223" xml:space="preserve">
On this page Harriot begins a general treatment of equations of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>d</mi><mi>a</mi><mi>a</mi></mstyle></math>,
with no square or linear term.
In order to preserve dimensions, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math> is used as a placeholder for a general 4-dimensional quantity.
All coeffcients are assumed to be positive.
Equations of this kind have two positive roots or none at all, depending on the size of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. <lb/>
For Harriot's derivation of the canonical form for unequal roots, see Add MS 6783, f. 173.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head163" xml:space="preserve" xml:lang="lat">
c.14.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1225" xml:space="preserve">
prob. 20. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex.
<lb/>[<emph style="it">tr:
Problem 20. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1226" xml:space="preserve">
Species canonica <lb/>
ad radices inæquales.
<lb/>[<emph style="it">tr:
Canonical form for unequal roots.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1227" xml:space="preserve">
Nam: Si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>. erit:
<lb/>[<emph style="it">tr:
For if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math> then:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1228" xml:space="preserve">
Et ita est:
<lb/>[<emph style="it">tr:
And so it is.
</emph>]<lb/>
</s>
<s xml:id="echoid-s1229" xml:space="preserve">
Si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math>. <lb/>
erit:
<lb/>[<emph style="it">tr:
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math> then:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1230" xml:space="preserve">
est igitur
<lb/>[<emph style="it">tr:
Therefore it is so.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1231" xml:space="preserve">
Species ad radices <lb/>
æquales.
<lb/>[<emph style="it">tr:
Canonical form for equal roots.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1232" xml:space="preserve">
Sunt continue proportionalia <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>
<lb/>[<emph style="it">tr:
There are continued proportionals <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1233" xml:space="preserve">
Sit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, minor radix. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, maior.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the smaller root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> the larger.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1234" xml:space="preserve">
fiat: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mo>,</mo><mn>3</mn><mo>:</mo><mi>d</mi><mo>,</mo><mfrac><mrow><mn>3</mn><mi>d</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>
<lb/>[<emph style="it">tr:
let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mo>:</mo><mn>3</mn><mo>=</mo><mi>d</mi><mo>:</mo><mfrac><mrow><mn>3</mn><mi>d</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1235" xml:space="preserve">
Dico quod:
<lb/>[<emph style="it">tr:
I say that
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1236" xml:space="preserve">
Est enim. est igitur.
<lb/>[<emph style="it">tr:
Indeed it is. Therefore it is so.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1237" xml:space="preserve">
Dico quod: <lb/>
vel:
<lb/>[<emph style="it">tr:
I say that: <lb/>
or:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1238" xml:space="preserve">
Est enim. est igitur.
<lb/>[<emph style="it">tr:
Indeed it is. Therefore it is so.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1239" xml:space="preserve">
Dico quod: <lb/>
vel:
<lb/>[<emph style="it">tr:
I say that: <lb/>
or:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1240" xml:space="preserve">
Est enim. <lb/>
Est igitur.
<lb/>[<emph style="it">tr:
Indeed it is. Therefore it is so.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1241" xml:space="preserve">
ergo. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>&gt;</mo><mi>a</mi></mstyle></math>. Hoc est qualibet <lb/>
radice.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>&gt;</mo><mi>a</mi></mstyle></math>. This is so whatever the root.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f404v" o="404v" n="809"/>
<pb file="add_6782_f405" o="405" n="810"/>
<div xml:id="echoid-div253" type="page_commentary" level="2" n="253">
<p>
<s xml:id="echoid-s1242" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1242" xml:space="preserve">
Following from the general treatment in Add MS 6782, f. 407, f. 406,
of avulsed quartics with no cube or square term,
Harriot here solves the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>1</mn><mn>7</mn><mn>9</mn><mn>4</mn><mn>4</mn><mo>=</mo><mn>2</mn><mn>7</mn><mn>7</mn><mn>5</mn><mn>5</mn><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> for both roots (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>8</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn><mn>7</mn></mstyle></math>).
He also shows how either root may be obtained from the other. <lb/>
The equation is taken from Problem 19 of Viète,
<emph style="it">De numerosa potestatum ad exegesin resolutione</emph>.
Viète gave rules for the relationship between the two roots but did not explain how he had arrived at them.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head164" xml:space="preserve" xml:lang="lat">
c.13.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1244" xml:space="preserve">
prob. 19. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex.
<lb/>[<emph style="it">tr:
Problem 19. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1245" xml:space="preserve">
Canon ad <lb/>
resolutionem
<lb/>[<emph style="it">tr:
Canonical form for the solution.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1246" xml:space="preserve">
Resolutio.
<lb/>[<emph style="it">tr:
Solution
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1247" xml:space="preserve">
Eductio radicis <lb/>
Minoris.
<lb/>[<emph style="it">tr:
Extraction of the smaller root.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1248" xml:space="preserve">
Radix igitur minor est 8.
<lb/>[<emph style="it">tr:
Therefore the smaller root is 8.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1249" xml:space="preserve">
Quæratur iam maior.
<lb/>[<emph style="it">tr:
Now the larger root is sought.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1250" xml:space="preserve">
Sit minor <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. maior <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the smaller root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> the larger.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1251" xml:space="preserve">
Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. 27.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>2</mn><mn>7</mn></mstyle></math> is given.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1252" xml:space="preserve">
Eductio radicis <lb/>
Maioris.
<lb/>[<emph style="it">tr:
Extraction of the larger root.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f405v" o="405v" n="811"/>
<pb file="add_6782_f406" o="406" n="812"/>
<div xml:id="echoid-div254" type="page_commentary" level="2" n="254">
<p>
<s xml:id="echoid-s1253" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1253" xml:space="preserve">
On this page, Harriot continues his general treatment of equations of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>,
begun on the previous page. <lb/>
The numerical example at the end, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>1</mn><mn>7</mn><mn>9</mn><mn>4</mn><mn>4</mn><mo>=</mo><mn>2</mn><mn>7</mn><mn>7</mn><mn>5</mn><mn>5</mn><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, is from Problem 19 of Viète,
<emph style="it">De numerosa potestatum ad exegesin resolutione</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head165" xml:space="preserve" xml:lang="lat">
c.12.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1255" xml:space="preserve">
prob. 19. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex.
<lb/>[<emph style="it">tr:
Problem 19. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1256" xml:space="preserve">
Species canonica <lb/>
ad radices inæquales.
<lb/>[<emph style="it">tr:
Canonical form for unequal roots.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1257" xml:space="preserve">
Si una radix sit nota, <lb/>
altera erit cognita.
<lb/>[<emph style="it">tr:
If one root is known, the other will be known.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1258" xml:space="preserve">
Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> nota. Quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is sought.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1259" xml:space="preserve">
Datur igitur, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1260" xml:space="preserve">
Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1261" xml:space="preserve">
Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. nota. Quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is sought.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1262" xml:space="preserve">
datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1263" xml:space="preserve">
Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1264" xml:space="preserve">
Pro exemplo ad resolutionem.
<lb/>[<emph style="it">tr:
According to this example for the solution.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1265" xml:space="preserve">
In numeris. Sit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. 8. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. 27
<lb/>[<emph style="it">tr:
In numbers, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mn>8</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>2</mn><mn>7</mn></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1266" xml:space="preserve">
Hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>1</mn><mn>7</mn><mo>,</mo><mn>9</mn><mn>4</mn><mn>4</mn><mo>=</mo><mn>2</mn><mn>7</mn><mo>,</mo><mn>7</mn><mn>5</mn><mn>5</mn><mo>,</mo><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
That is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>1</mn><mn>7</mn><mn>9</mn><mn>4</mn><mn>4</mn><mo>=</mo><mn>2</mn><mn>7</mn><mn>7</mn><mn>5</mn><mn>5</mn><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1267" xml:space="preserve">
Limites radicum.
<lb/>[<emph style="it">tr:
Limits of the roots.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f406v" o="406v" n="813"/>
<pb file="add_6782_f407" o="407" n="814"/>
<div xml:id="echoid-div255" type="page_commentary" level="2" n="255">
<p>
<s xml:id="echoid-s1268" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1268" xml:space="preserve">
On this page Harriot begins a general treatment of equations of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>d</mi><mi>a</mi><mi>a</mi></mstyle></math>,
with no cube or square term.
In order to preserve dimensions, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math> is used as a placeholder for a general 4-dimensional quantity;
similarly <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi><mi>d</mi></mstyle></math> is a placeholder for a 3-dimensional quantity, not necessarily a cube.
All coefficients are assumed to be positive.
Equations of this kind have two positive roots or none at all, depending on the size of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. <lb/>
For Harriot's derivation of the canonical form for unequal roots, see Add MS 6783, f. 174.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head166" xml:space="preserve" xml:lang="lat">
c.11.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1270" xml:space="preserve">
prob. 19. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex.
<lb/>[<emph style="it">tr:
Problem 19. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1271" xml:space="preserve">
Species canonica <lb/>
ad radices inæquales.
<lb/>[<emph style="it">tr:
Canonical form for unequal roots.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1272" xml:space="preserve">
nam: Si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>. erit:
<lb/>[<emph style="it">tr:
for if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>+</mo><mi>b</mi></mstyle></math> then:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1273" xml:space="preserve">
et ita est:
<lb/>[<emph style="it">tr:
and so it is.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1274" xml:space="preserve">
Si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math>. erit:
<lb/>[<emph style="it">tr:
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math> then:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1275" xml:space="preserve">
est enim.
<lb/>[<emph style="it">tr:
Indeed it is so.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1276" xml:space="preserve">
est igitur
<lb/>[<emph style="it">tr:
Therefore it is so
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1277" xml:space="preserve">
Species ad radices <lb/>
æquales.
<lb/>[<emph style="it">tr:
The case of equal roots.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1278" xml:space="preserve">
Sunt continue proportionalia. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>. <lb/>
et: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mi>c</mi><mo>,</mo><mi>b</mi><mi>b</mi><mi>c</mi><mi>c</mi><mo>,</mo><mi>b</mi><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
There are continued proportionals <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1279" xml:space="preserve">
Sit, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> minor radix. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, maior.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> the smaller root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> the larger.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1280" xml:space="preserve">
Dico etiam quod:
<lb/>[<emph style="it">tr:
I also say that:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1281" xml:space="preserve">
Est enim. Est igitur.
<lb/>[<emph style="it">tr:
Indeed it is. Therefore is is so.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1282" xml:space="preserve">
Ergo, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mo maxsize="1">[</mo></msqrt><mn>3</mn><mo maxsize="1">]</mo><mrow><mi>d</mi><mi>d</mi><mi>d</mi></mrow><mo>&gt;</mo><mi>a</mi></mstyle></math>. Hoc est qualibet radice.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mo maxsize="1">[</mo></msqrt><mn>3</mn><mo maxsize="1">]</mo><mrow><mi>d</mi><mi>d</mi><mi>d</mi></mrow><mo>&gt;</mo><mi>a</mi></mstyle></math>; this is so whatever the root.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f407v" o="407v" n="815"/>
<pb file="add_6782_f408" o="408" n="816"/>
<div xml:id="echoid-div256" type="page_commentary" level="2" n="256">
<p>
<s xml:id="echoid-s1283" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1283" xml:space="preserve">
Following from the general treatment in Add MS 6782, f. 410, f. 109, of avulsed cubics with no linear term,
Harriot here solves the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>4</mn><mn>3</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>5</mn><mn>7</mn><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> for both roots (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn><mn>0</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>4</mn><mn>5</mn></mstyle></math>).
He also shows how either root may be obtained from the other. <lb/>
The equation is taken from Problem 18 of Viète,
<emph style="it">De numerosa potestatum ad exegesin resolutione</emph>.
Viète gave rules for the relationship between the two roots but did not explain how he had arrived at them. <lb/>
This page demonstrates clearly two kinds of canonical forms used by Harriot in this treatise.
The first, the 'canonical form for unequal roots' is the general form of an avulsed cubic without a linear term;
in this case <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> are the two positive roots of the equation.
The second, the 'canonical form for the solution' is the form arrived at by assuming the root <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>
takes the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>0</mn><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>, that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is the first integer in the solution, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> the second.
Thus the meanings of <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> in the two forms are thus quite different.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head167" xml:space="preserve" xml:lang="lat">
c.10.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1285" xml:space="preserve">
prob. 18. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex.
<lb/>[<emph style="it">tr:
Problem 18. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1286" xml:space="preserve">
[<emph style="it">Note:
Here <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> are the two positive roots of the equation.
 </emph>]<lb/>
Species canonica <lb/>
ad radices inæquales.
<lb/>[<emph style="it">tr:
Canonical form for unequal roots.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1287" xml:space="preserve">
[<emph style="it">Note:
Here <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> are the first and second integers of any solution.
 </emph>]<lb/>
Species canonica <lb/>
ad resolutionem.
<lb/>[<emph style="it">tr:
Canonical form for the solution.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1288" xml:space="preserve">
Resolutio:
<lb/>[<emph style="it">tr:
Solution:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1289" xml:space="preserve">
Eductio lateris <lb/>
Minoris.
<lb/>[<emph style="it">tr:
Extraction of the smaller root.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1290" xml:space="preserve">
Radix igitur minor, est 30.
<lb/>[<emph style="it">tr:
Therefore the smaller root is 30.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1291" xml:space="preserve">
Quæratur iam maior.
<lb/>[<emph style="it">tr:
The larger root is now sought.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1292" xml:space="preserve">
Sit minor <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. maior <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let the smaller root be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, the larger <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1293" xml:space="preserve">
Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. 45.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>4</mn><mn>5</mn></mstyle></math> is given.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1294" xml:space="preserve">
Si quæratur minor radix. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
If the smaller root <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is sought,
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1295" xml:space="preserve">
Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. 30.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mn>3</mn><mn>0</mn></mstyle></math> is given.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1296" xml:space="preserve">
Eductio lateris <lb/>
Maioris.
<lb/>[<emph style="it">tr:
Solution and extraction of the larger root.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1297" xml:space="preserve">
Radix igitur maior 45.
<lb/>[<emph style="it">tr:
Therefore the larger root is 45.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f408v" o="408v" n="817"/>
<pb file="add_6782_f409" o="409" n="818"/>
<div xml:id="echoid-div257" type="page_commentary" level="2" n="257">
<p>
<s xml:id="echoid-s1298" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1298" xml:space="preserve">
On this page Harriot compares his own method with that of Viète, in Problem 18 of
<emph style="it">De numerosa potestatum ad exegesin resolutione</emph>,
showing that the two methods are essentially the same.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head168" xml:space="preserve" xml:lang="lat">
c.9.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1300" xml:space="preserve">
prob. 18. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex.
<lb/>[<emph style="it">tr:
Problem 18. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1301" xml:space="preserve">
Species canonica <lb/>
ad radices inæquales.
<lb/>[<emph style="it">tr:
Canonical form for unequal roots.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1302" xml:space="preserve">
Aliter quam supra.
<lb/>[<emph style="it">tr:
Another way from that above.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1303" xml:space="preserve">
si una radix sit nota, <lb/>
altera erit cognita.
<lb/>[<emph style="it">tr:
If one root is known, the other will be known.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1304" xml:space="preserve">
tres continue proportionales. <lb/>
prima et secunda <lb/>
secunda et tertia. <lb/>
tertia. <lb/>
prima.
<lb/>[<emph style="it">tr:
Three continued proportionals. <lb/>
first and second <lb/>
second and third <lb/>
third <lb/>
first
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1305" xml:space="preserve">
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> nota. Quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is sought.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1306" xml:space="preserve">
sint continue proportionales. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>e</mi><mi>e</mi></mrow><mrow><mi>f</mi></mrow></mfrac></mstyle></math>
<lb/>[<emph style="it">tr:
Let there be continued proportionals <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>e</mi><mi>e</mi></mrow><mrow><mi>f</mi></mrow></mfrac></mstyle></math>.]
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1307" xml:space="preserve">
Datur igitur. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1308" xml:space="preserve">
Datur igitur. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1309" xml:space="preserve">
Sed in Vieta <lb/>
iisdem præmissis.
<lb/>[<emph style="it">tr:
But in Viète from the same premises.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1310" xml:space="preserve">
[<emph style="it">Note:
The other sheet referred to here is Add MS 6782, f. 410.
 </emph>]<lb/>
eadem quæ altera charta.
<lb/>[<emph style="it">tr:
The same as in the other sheet.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1311" xml:space="preserve">
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> nota. Quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is sought.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1312" xml:space="preserve">
Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1313" xml:space="preserve">
datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1314" xml:space="preserve">
sed in Vieta
<lb/>[<emph style="it">tr:
But in Viete
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1315" xml:space="preserve">
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi><mo>=</mo><mo>-</mo><mi>g</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>b</mi></mstyle></math>. <lb/>
eadem quæ nostra in altera charta.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi><mo>=</mo><mo>-</mo><mi>g</mi><mi>b</mi><mo>=</mo><mi>b</mi><mi>b</mi></mstyle></math>, the same as mine in the other sheet.
</emph>]<lb/>
[<emph style="it">Note:
The other sheet referred to here is Add MS 6782, f. 410.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6782_f409v" o="409v" n="819"/>
<pb file="add_6782_f410" o="410" n="820"/>
<div xml:id="echoid-div258" type="page_commentary" level="2" n="258">
<p>
<s xml:id="echoid-s1316" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1316" xml:space="preserve">
On this page, Harriot continues his general treatment of equations of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>,
begun on the previous page. <lb/>
The numerical example at the end, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>4</mn><mn>3</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>5</mn><mn>7</mn><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, is from Problem 18 of Viète,
<emph style="it">De numerosa potestatum ad exegesin resolutione</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head169" xml:space="preserve" xml:lang="lat">
c.8.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1318" xml:space="preserve">
prob. 18. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex.
<lb/>[<emph style="it">tr:
Problem 18. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1319" xml:space="preserve">
Species canonica <lb/>
ad radices inæquales.
<lb/>[<emph style="it">tr:
Canonical form for unequal roots.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1320" xml:space="preserve">
Si una radix sit nota, <lb/>
altera erit cognita.
<lb/>[<emph style="it">tr:
If one root is known, the other will be known.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1321" xml:space="preserve">
</s>
<lb/>
<s xml:id="echoid-s1322" xml:space="preserve">
Dabitur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> will be given.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1323" xml:space="preserve">
Datur igitur, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1324" xml:space="preserve">
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> nota. Quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is sought.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1325" xml:space="preserve">
Dabitur ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> will be given.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1326" xml:space="preserve">
Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1327" xml:space="preserve">
pro exemplo ad resolutionem.
<lb/>[<emph style="it">tr:
According to this example for the solution.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1328" xml:space="preserve">
In numeris sit. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. 30. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. 45.
<lb/>[<emph style="it">tr:
In numbers, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mn>3</mn><mn>0</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>4</mn><mn>5</mn></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1329" xml:space="preserve">
Hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>4</mn><mn>3</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>5</mn><mn>7</mn><mo>,</mo><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
That is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>4</mn><mn>3</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>5</mn><mn>7</mn><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s1330" xml:space="preserve">
Limites radicum.
<lb/>[<emph style="it">tr:
Limits of the roots.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f410v" o="410v" n="821"/>
<pb file="add_6782_f411" o="411" n="822"/>
<div xml:id="echoid-div259" type="page_commentary" level="2" n="259">
<p>
<s xml:id="echoid-s1331" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1331" xml:space="preserve">
On this page Harriot begins a general treatment of equations of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>,
with no linear term. All coefficients are assumed to be positive.
Equations of this kind have two positive roots or none at all, depending on the size of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. <lb/>
For Harriot's derivation of the canonical form for unequal roots, see Add MS 6783, f. 181.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head170" xml:space="preserve" xml:lang="lat">
c.7.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1333" xml:space="preserve">
prob. 18. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex.
<lb/>[<emph style="it">tr:
Problem 18. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1334" xml:space="preserve">
Species canonica <lb/>
ad radices inæquales.
<lb/>[<emph style="it">tr:
Canonical form for unequal roots.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1335" xml:space="preserve">
Nam: Si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>. <lb/>
erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>b</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mi>b</mi><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>. Et ita est:
<lb/>[<emph style="it">tr:
For if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>b</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mi>b</mi><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>; and so it is.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1336" xml:space="preserve">
Si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math>. <lb/>
erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mi>c</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>c</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>c</mi><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>. est enim.
<lb/>[<emph style="it">tr:
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mi>c</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>c</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>c</mi><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>; indeed it is.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1337" xml:space="preserve">
est igitur
<lb/>[<emph style="it">tr:
Therefore it is so.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1338" xml:space="preserve">
Species ad radices <lb/>
æquales.
<lb/>[<emph style="it">tr:
The case of equal roots.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1339" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math> sunt continue proportionalia.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math> are in continued proportion.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1340" xml:space="preserve">
Dico etiam quod:
<lb/>[<emph style="it">tr:
I say also that:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1341" xml:space="preserve">
ponatur: est igitur:
<lb/>[<emph style="it">tr:
This supposed, then
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1342" xml:space="preserve">
est enim, est igitur.
<lb/>[<emph style="it">tr:
Indded it is; therefore it is so.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1343" xml:space="preserve">
est igitur: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>&gt;</mo><mi>a</mi></mstyle></math>. hoc est qualibet <lb/>
radice.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>&gt;</mo><mi>a</mi></mstyle></math>; this is so whatever the root.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f411v" o="411v" n="823"/>
<pb file="add_6782_f412" o="412" n="824"/>
<div xml:id="echoid-div260" type="page_commentary" level="2" n="260">
<p>
<s xml:id="echoid-s1344" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1344" xml:space="preserve">
Following on from Add MS 6782, f. 415, f. 414, f. 413, Harriot here solves the equation
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>5</mn><mn>5</mn><mn>5</mn><mn>2</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>3</mn><mn>1</mn><mn>0</mn><mn>4</mn><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> for the larger root (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>1</mn><mn>0</mn><mn>8</mn></mstyle></math>).
He then shows how the smaller roots (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>1</mn><mn>2</mn></mstyle></math>) may be derived from the larger one.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head171" xml:space="preserve" xml:lang="lat">
c.6.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1346" xml:space="preserve">
prob. 17. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex.
<lb/>[<emph style="it">tr:
Problem 17. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1347" xml:space="preserve">
Species canonica <lb/>
ad resolutionem.
<lb/>[<emph style="it">tr:
Canonical form for the solution.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1348" xml:space="preserve">
Resolutio <lb/>
et eductio lateris <lb/>
Maioris.
<lb/>[<emph style="it">tr:
Solution and extraction of the larger root.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1349" xml:space="preserve">
Radix <emph style="super">igitur</emph> maior est 108.
<lb/>[<emph style="it">tr:
Therefore the larger root is 108.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1350" xml:space="preserve">
Quæratur iam minor.
<lb/>[<emph style="it">tr:
Now there is sought the smaller root.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1351" xml:space="preserve">
Sit maior <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. minor <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let the larger root be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, the smaller <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1352" xml:space="preserve">
Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. 12.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mn>1</mn><mn>2</mn></mstyle></math> is given.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f412v" o="412v" n="825"/>
<pb file="add_6782_f413" o="413" n="826"/>
<div xml:id="echoid-div261" type="page_commentary" level="2" n="261">
<p>
<s xml:id="echoid-s1353" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1353" xml:space="preserve">
Following from the general treatment in Add MS 6782, f. 415, f. 414, of avulsed cubics with no square term,
Harriot here solves the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>5</mn><mn>5</mn><mn>5</mn><mn>2</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>3</mn><mn>1</mn><mn>0</mn><mn>4</mn><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> for the smaller root (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>1</mn><mn>2</mn></mstyle></math>).
He then shows how the larger roots (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>1</mn><mn>0</mn><mn>8</mn></mstyle></math>) may be derived from the smaller one. <lb/>
The equation is taken from Problem 17 of Viète,
<emph style="it">De numerosa potestatum ad exegesin resolutione</emph>.
Viète gave rules for the relationship between the two roots but did not explain how he had arrived at them.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head172" xml:space="preserve" xml:lang="lat">
c.5.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1355" xml:space="preserve">
prob. 17. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex.
<lb/>[<emph style="it">tr:
Problem 17. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1356" xml:space="preserve">
Species canonica <lb/>
ad resolutionem.
<lb/>[<emph style="it">tr:
Canonical form for the solution.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1357" xml:space="preserve">
Resolutio <lb/>
et eductio lateris <lb/>
minoris.
<lb/>[<emph style="it">tr:
Solution and extraction of the smaller root.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1358" xml:space="preserve">
Radix igitur minor est, 12.
<lb/>[<emph style="it">tr:
Therefore the smaller root is 12.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1359" xml:space="preserve">
Radix <emph style="super">minor</emph> sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. maior <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. <lb/>
Et quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
The smaller root is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, the larger <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>; and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is sought.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1360" xml:space="preserve">
Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. 108.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>1</mn><mn>0</mn><mn>8</mn></mstyle></math> is given.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1361" xml:space="preserve">
A. Poristicum
<lb/>[<emph style="it">tr:
Proof
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1362" xml:space="preserve">
Quod: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi><mo>-</mo><mi>b</mi><mi>b</mi><mo>=</mo><mfrac><mrow><mi>x</mi><mi>x</mi><mi>z</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>
<lb/>[<emph style="it">tr:
Because <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi><mo>-</mo><mi>b</mi><mi>b</mi><mo>=</mo><mfrac><mrow><mi>x</mi><mi>x</mi><mi>z</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1363" xml:space="preserve">
Hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
That is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1364" xml:space="preserve">
Est enim. est igitur.
<lb/>[<emph style="it">tr:
Indeed it is; therefore it is so.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1365" xml:space="preserve">
Etiam. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi><mo>-</mo><mi>c</mi><mi>c</mi><mo>=</mo><mfrac><mrow><mi>x</mi><mi>x</mi><mi>z</mi></mrow><mrow><mi>c</mi></mrow></mfrac></mstyle></math>.
<lb/>[<emph style="it">tr:
Also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi><mo>-</mo><mi>c</mi><mi>c</mi><mo>=</mo><mfrac><mrow><mi>x</mi><mi>x</mi><mi>z</mi></mrow><mrow><mi>c</mi></mrow></mfrac></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1366" xml:space="preserve">
hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
That is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1367" xml:space="preserve">
Est enim. est igitur.
<lb/>[<emph style="it">tr:
Indeed it is; therefore it is so.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f413v" o="413v" n="827"/>
<pb file="add_6782_f414" o="414" n="828"/>
<div xml:id="echoid-div262" type="page_commentary" level="2" n="262">
<p>
<s xml:id="echoid-s1368" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1368" xml:space="preserve">
On this page, Harriot continues his general treatment of equations of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>,
begun on the previous page. <lb/>
The numerical example at the end, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>5</mn><mn>5</mn><mn>5</mn><mn>2</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>3</mn><mn>1</mn><mn>0</mn><mn>4</mn><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, is from Problem 17 of Viète,
<emph style="it">De numerosa potestatum ad exegesin resolutione</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head173" xml:space="preserve" xml:lang="lat">
c.4.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1370" xml:space="preserve">
prob. 17. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex.
<lb/>[<emph style="it">tr:
Problem 17. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1371" xml:space="preserve">
Species canonica <lb/>
ad radices inæquales.
<lb/>[<emph style="it">tr:
Canonical form for unequal roots.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1372" xml:space="preserve">
Si una radix sit nota, <lb/>
altera erit cognita.
<lb/>[<emph style="it">tr:
If one root is known, the other will be known.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1373" xml:space="preserve">
Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> nota. Quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is sought.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1374" xml:space="preserve">
Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1375" xml:space="preserve">
vel:
<lb/>[<emph style="it">tr:
or:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1376" xml:space="preserve">
Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1377" xml:space="preserve">
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> nota. Quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is sought.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1378" xml:space="preserve">
Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1379" xml:space="preserve">
vel:
<lb/>[<emph style="it">tr:
or:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1380" xml:space="preserve">
Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1381" xml:space="preserve">
Pro exemplo ad resolutionem.
<lb/>[<emph style="it">tr:
According to this example for the solution.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1382" xml:space="preserve">
In numeris. Sit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. 12. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. 108.
<lb/>[<emph style="it">tr:
In numbers, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mn>1</mn><mn>2</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>1</mn><mn>0</mn><mn>8</mn></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1383" xml:space="preserve">
Limites radicum.
<lb/>[<emph style="it">tr:
Limits of the roots.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f414v" o="414v" n="829"/>
<pb file="add_6782_f415" o="415" n="830"/>
<div xml:id="echoid-div263" type="page_commentary" level="2" n="263">
<p>
<s xml:id="echoid-s1384" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1384" xml:space="preserve">
On this page Harriot begins a general treatment of equations of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>,
with no square term. To preserve dimensions, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math> is used as a placeholder for a general 3-dimensional quantity;
similarly <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi></mstyle></math> is a placeholder for a 2-dimensional quantity, not necessarily a square.
All coefficients are assumed to be positive.
Equations of this kind have two positive roots or none at all, depending on the size of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. <lb/>
For Harriot's derivation of the canonical form for unequal roots, see Add MS 6783, f. 181.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head174" xml:space="preserve" xml:lang="lat">
c.3.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1386" xml:space="preserve">
prob. 17. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex.
<lb/>[<emph style="it">tr:
Problem 17. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1387" xml:space="preserve">
Species canonica <lb/>
ad radices inæquales.
<lb/>[<emph style="it">tr:
Canonical form for unequal roots.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1388" xml:space="preserve">
Nam: Si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>. erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>b</mi><mi>b</mi></mstyle></math>. Et ita est:
<lb/>[<emph style="it">tr:
For if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>b</mi><mi>b</mi></mstyle></math>; and so it is.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1389" xml:space="preserve">
Si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math>. erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>. est enim.
<lb/>[<emph style="it">tr:
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>; and indeed it is so.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1390" xml:space="preserve">
est igitur <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore it is so, that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1391" xml:space="preserve">
Species ad radices <lb/>
æquales.
<lb/>[<emph style="it">tr:
The case of equal roots.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1392" xml:space="preserve">
si: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math> <lb/>
erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>.
</emph>]<lb/>
[<emph style="it">Note:
If the cubic has two positive roots <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>, then the third root must be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mi>b</mi><mo>-</mo><mi>c</mi></mstyle></math>.
Hence the product of the roots is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mo maxsize="1">(</mo><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo maxsize="1">)</mo></mstyle></math>.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1393" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math>. sunt continue <lb/>
proportionalia.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math>, are in continued proportion.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1394" xml:space="preserve">
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, minor radix. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, maior.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the smaller root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> the larger.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1395" xml:space="preserve">
Ergo qualibet radix non <lb/>
habet plures figuras <lb/>
quam sunt in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>d</mi><mi>d</mi></mrow></msqrt></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore whatever the root, it has no more figures than are in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>d</mi><mi>d</mi></mrow></msqrt></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f415v" o="415v" n="831"/>
<pb file="add_6782_f416" o="416" n="832"/>
<div xml:id="echoid-div264" type="page_commentary" level="2" n="264">
<p>
<s xml:id="echoid-s1396" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1396" xml:space="preserve">
On the previous page, Add MS 6782, f. 417, Harriot discussed equations of the general form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>.
On that page, as an example, he calculated limits for the two positive roots of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>9</mn><mn>2</mn><mn>6</mn><mn>1</mn><mo>=</mo><mn>3</mn><mn>7</mn><mn>0</mn><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>.
On this page he solves the same equation fully for both roots. <lb/>
The equation is taken from Problem 16 of Viète,
<emph style="it">De numerosa potestatum ad exegesin resolutione</emph>.
Viète gave rules for the relationship between the two roots but did not explain how he had arrived at them.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head175" xml:space="preserve" xml:lang="lat">
c.2.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1398" xml:space="preserve">
prob. 16. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex.
<lb/>[<emph style="it">tr:
Problem 16. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1399" xml:space="preserve">
Species canonica <lb/>
ad resolutione.
<lb/>[<emph style="it">tr:
Canonical form for the solution.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1400" xml:space="preserve">
Resolutio, <lb/>
et eductio <lb/>
lateris minoris
<lb/>[<emph style="it">tr:
Solution and extraction of the smaller root.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1401" xml:space="preserve">
Divisor
<lb/>[<emph style="it">tr:
Divisor
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1402" xml:space="preserve">
Ergo. 27. latus minus. <lb/>
ergo: latus maius. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mn>7</mn><mn>0</mn><mo>-</mo><mn>2</mn><mn>7</mn><mo>=</mo><mn>3</mn><mn>4</mn><mn>3</mn></mstyle></math>. <lb/>
vel: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>9</mn><mn>2</mn><mn>6</mn><mn>1</mn></mrow><mrow><mn>2</mn><mn>7</mn></mrow></mfrac><mo>=</mo><mn>3</mn><mn>4</mn><mn>3</mn></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore 27 is the smaller root. <lb/>
Therefore the larger root is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mn>7</mn><mn>0</mn><mo>-</mo><mn>2</mn><mn>7</mn><mo>=</mo><mn>3</mn><mn>4</mn><mn>3</mn></mstyle></math>. <lb/>
or: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>9</mn><mn>2</mn><mn>6</mn><mn>1</mn></mrow><mrow><mn>2</mn><mn>7</mn></mrow></mfrac><mo>=</mo><mn>3</mn><mn>4</mn><mn>3</mn></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1403" xml:space="preserve">
Eductio <lb/>
lateris maioris.
<lb/>[<emph style="it">tr:
Extraction of the larger root.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1404" xml:space="preserve">
Divisor.
<lb/>[<emph style="it">tr:
Divisor.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1405" xml:space="preserve">
Divisor.
<lb/>[<emph style="it">tr:
Divisor.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1406" xml:space="preserve">
Ergo. latus maius. 343 <lb/>
ergo laus minor. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mn>7</mn><mn>0</mn><mo>-</mo><mn>3</mn><mn>4</mn><mn>3</mn><mo>=</mo><mn>2</mn><mn>7</mn></mstyle></math>. <lb/>
vel. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>9</mn><mn>2</mn><mn>6</mn><mn>1</mn></mrow><mrow><mn>3</mn><mn>4</mn><mn>3</mn></mrow></mfrac><mo>=</mo><mn>2</mn><mn>7</mn></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore the greater root is 343. <lb/>
therefore the smaller root is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mn>7</mn><mn>0</mn><mo>-</mo><mn>3</mn><mn>4</mn><mn>3</mn><mo>=</mo><mn>2</mn><mn>7</mn></mstyle></math>. <lb/>
or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>9</mn><mn>2</mn><mn>6</mn><mn>1</mn></mrow><mrow><mn>3</mn><mn>4</mn><mn>3</mn></mrow></mfrac><mo>=</mo><mn>2</mn><mn>7</mn></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f416v" o="416v" n="833"/>
<pb file="add_6782_f417" o="417" n="834"/>
<div xml:id="echoid-div265" type="page_commentary" level="2" n="265">
<p>
<s xml:id="echoid-s1407" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1407" xml:space="preserve">
This is the first of a set of 18 pages on extracting roots of avulsed equations,
that is, equations containing three terms, in which the term of highest degree is subtracted
(torn away, or 'avulsed') from the term of next higest degree.
Such equations have two positive roots. It is therefore important to know the relative sizes of the roots
before beginning extraction by numerical methods. This is the problem Harriot investigates in this section. <lb/>
The work is closely based on Problems 16 to 20 in
Viète, <emph style="it">De numerosa potestatum ad exegesin resolutione</emph>, 1600.
Viète gave rules for finding the second positive root once the first is known, but without explanation.
In this section, Harriot fills in the missing details, showing how the two positive roots
are related to the coefficients of the original equation and to each other. <lb/>
For a general explanation of the method of extraction see Add MS 6782, f. 399.
For further discussion see See Stedall 2003, 87–123 and 294, and Stedall 2011, 29–33. <lb/>
This first page of Section c is transcribed in full, but for subsequent pages,
only phrases and sentences are transcribed, not the calculations or the single words used in them. <lb/>
For another version of the first page see Add MS 6783, f. 62v, f. 62. <lb/>
For Harriot's derivation of the canonical form for unequal roots, see Add MS 6783, f. 183.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head176" xml:space="preserve" xml:lang="lat">
c.1.) De numerosa potestatum resolutione.
<lb/>[<emph style="it">tr:
On the numerical resolution of powers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1409" xml:space="preserve">
prob. 16. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex.
<lb/>[<emph style="it">tr:
Problem 16. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1410" xml:space="preserve">
Species canonica <lb/>
ad radices inæquales.
<lb/>[<emph style="it">tr:
Canonical form for unequal roots.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1411" xml:space="preserve">
Nam: Si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>. erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>b</mi></mstyle></math>. et ita est.
<lb/>[<emph style="it">tr:
For if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>b</mi></mstyle></math>; and so it is.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1412" xml:space="preserve">
Si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math>. erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mo>-</mo><mi>c</mi><mi>c</mi></mstyle></math>. est enim.
<lb/>[<emph style="it">tr:
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mo>-</mo><mi>c</mi><mi>c</mi></mstyle></math>; indeed it is so.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1413" xml:space="preserve">
est igitur <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
therefore it is so, that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1414" xml:space="preserve">
Species ad radices <lb/>
æquales. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>=</mo><mi>b</mi><mi>a</mi><mo>+</mo><mi>b</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>
<lb/>[<emph style="it">tr:
Canonical form for equal roots.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1415" xml:space="preserve">
Vel: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>=</mo><mn>2</mn><mi>b</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>=</mo><mn>2</mn><mi>b</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1416" xml:space="preserve">
si: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi><mo>=</mo><mi>d</mi></mstyle></math> <lb/>
erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>=</mo><mi>x</mi><mi>z</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
if: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi><mo>=</mo><mi>d</mi></mstyle></math> <lb/>
then: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>=</mo><mi>x</mi><mi>z</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1417" xml:space="preserve">
Sunt in ratione inæqualitatis: <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>.
<lb/>[<emph style="it">tr:
Let <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>, be in unequal ratio.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1418" xml:space="preserve">
sit, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> radix minor. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, maior. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>b</mi><mo>&lt;</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>&lt;</mo><mn>2</mn><mi>c</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>&lt;</mo><mfrac><mrow><mi>b</mi><mo>+</mo><mi>c</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>&lt;</mo><mi>c</mi></mstyle></math>. <lb/>
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>&lt;</mo><mfrac><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>&lt;</mo><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the smaller root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> the larger. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>b</mi><mo>&lt;</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>&lt;</mo><mn>2</mn><mi>c</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>&lt;</mo><mfrac><mrow><mi>b</mi><mo>+</mo><mi>c</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>&lt;</mo><mi>c</mi></mstyle></math>. <lb/>
Therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>&lt;</mo><mfrac><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>&lt;</mo><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1419" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>&lt;</mo><mi>b</mi><mi>c</mi><mo>&lt;</mo><mi>c</mi><mi>c</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>&lt;</mo><msqrt><mrow><mi>b</mi><mi>c</mi></mrow></msqrt><mo>&lt;</mo><mi>c</mi></mstyle></math>. <lb/>
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>&lt;</mo><msqrt><mrow><mi>x</mi><mi>z</mi></mrow></msqrt><mo>&lt;</mo><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>&lt;</mo><mi>b</mi><mi>c</mi><mo>&lt;</mo><mi>c</mi><mi>c</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>&lt;</mo><msqrt><mrow><mi>b</mi><mi>c</mi></mrow></msqrt><mo>&lt;</mo><mi>c</mi></mstyle></math>. <lb/>
Therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>&lt;</mo><msqrt><mrow><mi>x</mi><mi>z</mi></mrow></msqrt><mo>&lt;</mo><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1420" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>,</mo><mn>2</mn><mo>,</mo><mi>x</mi><mi>z</mi><mo>:</mo><mn>1</mn><mo>,</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mi>z</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math> <lb/>
Dico quod: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>&lt;</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mi>z</mi></mrow><mrow><mi>d</mi></mrow></mfrac><mo>&lt;</mo><mi>c</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi><mo>&lt;</mo><mn>2</mn><mo>,</mo><mi>x</mi><mi>z</mi><mo>&lt;</mo><mi>c</mi><mi>d</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>&lt;</mo><mn>2</mn><mi>b</mi><mi>c</mi><mo>&lt;</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi></mstyle></math> <lb/>
est enim. est igitur
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>:</mo><mn>2</mn><mi>x</mi><mi>z</mi><mo>=</mo><mn>1</mn><mo>:</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mi>z</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math> <lb/>
I say that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>&lt;</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mi>z</mi></mrow><mrow><mi>d</mi></mrow></mfrac><mo>&lt;</mo><mi>c</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi><mo>&lt;</mo><mn>2</mn><mi>x</mi><mi>z</mi><mo>&lt;</mo><mi>c</mi><mi>d</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>&lt;</mo><mn>2</mn><mi>b</mi><mi>c</mi><mo>&lt;</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi></mstyle></math> <lb/>
indeed it is; therefore it is so.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1421" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math> <lb/>
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>a</mi><mo>&gt;</mo><mi>a</mi><mi>a</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>&gt;</mo><mi>a</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math> <lb/>
Therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>a</mi><mo>&gt;</mo><mi>a</mi><mi>a</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>&gt;</mo><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
<s xml:id="echoid-s1422" xml:space="preserve">
Hoc est qualibet <lb/>
radice.
<lb/>[<emph style="it">tr:
This is so whatever the root.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1423" xml:space="preserve">
Ergo [???] radix habet plures <lb/>
figuras quam sunt in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore the [???] root has more figures than are in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1424" xml:space="preserve">
Si una radix sit nota: <lb/>
altera erit cognita.
<lb/>[<emph style="it">tr:
If one root is known, the other will be known.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1425" xml:space="preserve">
Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> nota. quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is sought.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1426" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>=</mo><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> <lb/>
ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>b</mi><mo>=</mo><mi>c</mi></mstyle></math>. <lb/>
vel: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi><mo>=</mo><mi>b</mi><mi>c</mi></mstyle></math> <lb/>
ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>x</mi><mi>z</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>=</mo><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> <lb/>
therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>b</mi><mo>=</mo><mi>c</mi></mstyle></math>. <lb/>
or: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi><mo>=</mo><mi>b</mi><mi>c</mi></mstyle></math> <lb/>
therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>x</mi><mi>z</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1427" xml:space="preserve">
Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> nota. quæratur, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>c</mi><mo>=</mo><mi>b</mi></mstyle></math> <lb/>
et: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>x</mi><mi>z</mi></mrow><mrow><mi>c</mi></mrow></mfrac><mo>=</mo><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is sought. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>c</mi><mo>=</mo><mi>b</mi></mstyle></math> <lb/>
and: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>x</mi><mi>z</mi></mrow><mrow><mi>c</mi></mrow></mfrac><mo>=</mo><mi>b</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1428" xml:space="preserve">
Pro exemplo ad resolutionem.
<lb/>[<emph style="it">tr:
According to this example for the solution.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1429" xml:space="preserve">
In numeris sit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. 27. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> 343.
<lb/>[<emph style="it">tr:
In numbers, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mn>2</mn><mn>7</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>3</mn><mn>4</mn><mn>3</mn></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1430" xml:space="preserve">
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>7</mn><mo>×</mo><mn>3</mn><mn>4</mn><mn>3</mn><mo>=</mo><mn>2</mn><mn>7</mn><mo>,</mo><mi>a</mi><mo>+</mo><mn>3</mn><mn>4</mn><mn>3</mn><mo>,</mo><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>. <lb/>
Hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>9</mn><mn>2</mn><mn>6</mn><mn>1</mn><mo>=</mo><mn>3</mn><mn>7</mn><mn>0</mn><mo>,</mo><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>7</mn><mo>×</mo><mn>3</mn><mn>4</mn><mn>3</mn><mo>=</mo><mn>2</mn><mn>7</mn><mi>a</mi><mo>+</mo><mn>3</mn><mn>4</mn><mn>3</mn><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>. <lb/>
That is: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>9</mn><mn>2</mn><mn>6</mn><mn>1</mn><mo>=</mo><mn>3</mn><mn>7</mn><mn>0</mn><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1431" xml:space="preserve">
Limites radicum ex præcedentibus.
<lb/>[<emph style="it">tr:
The limits of the roots from what has gone before.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f417v" o="417v" n="835"/>
<p>
<s xml:id="echoid-s1432" xml:space="preserve">
Johan Dycker <lb/>
Johan
</s>
</p>
<pb file="add_6782_f418" o="418" n="836"/>
<pb file="add_6782_f418v" o="418v" n="837"/>
<pb file="add_6782_f419" o="419" n="838"/>
<div xml:id="echoid-div266" type="page_commentary" level="2" n="266">
<p>
<s xml:id="echoid-s1433" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1433" xml:space="preserve">The referenceon this page is to Proposition 15 from Chapter 19 of Viète's
<emph style="it">Variorum responsorum liber VIII</emph> (1593).
</s>
<lb/>
<quote xml:lang="lat">
XV. Datis tribus lateribus, dantur anguli.
</quote>
<lb/>
<quote>
Given three sides, the angles are given.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head177" xml:space="preserve" xml:lang="lat">
Lemmata quædam <lb/>
ad praxin prop. 15 <lb/>
Vieta lib. resp. 8. <lb/>
pag. 35
<lb/>[<emph style="it">tr:
Certain lemmas for carrying out Proposition 15, Viète, Responsorum liber VIII, page 35.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1435" xml:space="preserve">
1. Duæ peripheriæ sigillatim minores quadranti, et earum complementa: <lb/>
æqualem habent differentiam. <lb/>
sint duæ peripheriæ <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>, differentia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> <lb/>
complementa earum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>o</mi></mstyle></math>, differentia etiam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
1. Two arcs each less than a quadrant, and their complements, have equal differences. <lb/>
Let the two arcs be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> abd <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>, with difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. <lb/>
Their complements are <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi></mstyle></math> adn <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>o</mi></mstyle></math>, also with difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1436" xml:space="preserve">
2. Complementa aggregati duarum peripheriarum sigillatim minorum quadranti; et <lb/>
aggregatum illarum complementorum: sunt æqualia. <lb/>
aggregatum ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math>. complementum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>d</mi></mstyle></math>. <lb/>
complementum peripheriæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi></mstyle></math>, hoc est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>o</mi></mstyle></math>. <lb/>
complementum peripheriæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>, est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>o</mi></mstyle></math>. <lb/>
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>o</mi><mo>+</mo><mi>o</mi><mi>c</mi></mstyle></math> est aggregatum complementorum <lb/>
<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><mi>h</mi><mi>i</mi></mstyle></math> sunt æquales. <lb/>
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>o</mi><mi>d</mi></mstyle></math> est aggregatum complementorum et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>d</mi></mstyle></math> complementum aggregat.
<lb/>[<emph style="it">tr:
2. The complements of the sum of two arcs each less than a quadrant,
and the sum of those complements, are equal. <lb/>
Therefore if the sum is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math> the complement is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>d</mi></mstyle></math>. <lb/>
The complement of the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi></mstyle></math>, that is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>o</mi></mstyle></math>. <lb/>
The complement of the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>o</mi></mstyle></math>. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>o</mi><mo>+</mo><mi>o</mi><mi>c</mi></mstyle></math> is the sum of the complements. <lb/>
<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>h</mi><mi>i</mi></mstyle></math> are equal. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>o</mi><mo>+</mo><mi>o</mi><mi>c</mi></mstyle></math> is the sum of the complements. <lb/>
<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>h</mi><mi>i</mi></mstyle></math> are equal. <lb/>
Therfore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>o</mi><mi>d</mi></mstyle></math> is the sum of the coplements and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>d</mi></mstyle></math> is the complement of the sum.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f419v" o="419v" n="839"/>
<pb file="add_6782_f420" o="420" n="840"/>
<div xml:id="echoid-div267" type="page_commentary" level="2" n="267">
<p>
<s xml:id="echoid-s1437" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1437" xml:space="preserve">
This page continues Harriot's work from Add MS 6787, f. 61, and Add MS 782, f. 422.
on Viète's statement of 'Syntomon'. <lb/>
The third case is where one angle is greater than a right angle, the other less.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head178" xml:space="preserve" xml:lang="lat">
<foreign xml:lang="gre">Syntomon</foreign> Secundo.
<lb/>[<emph style="it">tr:
Syntomon, third case.
</emph>]<lb/>
</head>
<pb file="add_6782_f420v" o="420v" n="841"/>
<pb file="add_6782_f421" o="421" n="842"/>
<div xml:id="echoid-div268" type="page_commentary" level="2" n="268">
<p>
<s xml:id="echoid-s1439" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1439" xml:space="preserve">
This page continues Harriot's work from Add MS 6787, f. 61,
on Viète's statement of 'Syntomon'. <lb/>
The second case is where both angles are greater than a right angle.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head179" xml:space="preserve" xml:lang="lat">
Vieta lib. 8. resp. <lb/>
pag. 39. <lb/>
<foreign xml:lang="gre">Syntomon</foreign> Secundo.
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 39, Syntomon, second case.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1441" xml:space="preserve">
Interpretatio. <lb/>
<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 <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi><mo>=</mo><mi>a</mi><mi>d</mi></mstyle></math> <lb/>
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math> differentia.
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1442" xml:space="preserve">
1. Si duo rectangula fuerint sigillatim applicata ad <lb/>
2. sinum totum; unum, <emph style="super">*</emph> <emph style="ul">duorum sinum</emph>,
quorum utraque peripheriæ <lb/>
sunt quadranti minores; alterum <emph style="super">+</emph> <emph style="ul">Maiorem periphe-</emph> <lb/>
<emph style="ul">riæ sinum</emph> complementarum: Duæ latitudines <lb/>
oriundæ component sinum complementi differentiæ <lb/>
peripheriarum.
<lb/>[<emph style="it">tr:
1, 2. If two rectangles are each applied to the whole sine,
one <emph style="super">*</emph>of two sines, of which either arc is less than the quadrant,
the other <emph style="super">+</emph> greater than the sine of the complement of the arc,
then the two latitudes arising are composed of the sine of the complement of the differences of the arcs.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1443" xml:space="preserve">
* sub duobus sinibus
<lb/>[<emph style="it">tr:
* under two sines
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1444" xml:space="preserve">
+ sub illarum <lb/>
sinibus
<lb/>[<emph style="it">tr:
under the sines of them
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1445" xml:space="preserve">
3. Si duo rectangula fuerint sigillatimm applicata ad <lb/>
4. sinum totum; unum, <emph style="super">*</emph> <emph style="ul">duorum sinum</emph>,
quorum peripheriæ sunt <lb/>
affectionis inter se diversæ; alterum, <emph style="ul">illarum peripheriæ</emph> <lb/>
<emph style="ul">sinum</emph> complementarum: Duæ latitudines oriundæ <lb/>
component sinum complementi aggregati peripheriæ.
<lb/>[<emph style="it">tr:
3, 4. If two rectangles are each applied to the whole sine,
one <emph style="super">*</emph>of two sines, of which the relationship to the arc is different,
the other the complements of the sines of those arcs,
then the two latitudes arising are composed of the sine of the complement of the sum of the arcs.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f421v" o="421v" n="843"/>
<pb file="add_6782_f422" o="422" n="844"/>
<div xml:id="echoid-div269" type="page_commentary" level="2" n="269">
<p>
<s xml:id="echoid-s1446" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1446" xml:space="preserve">The reference on this page is to Proposition 20 from Chapter 19 of Viète's
<emph style="it">Variorum responsorum liber VIII</emph> (1593).
</s>
<lb/>
<quote xml:lang="lat">
XX. <lb/>
Trianguli cujuslibet sphærici. <lb/>
Datis angulis duobus, &amp; latere quod iis adjacent, datur angulus reliquus.
</quote>
<lb/>
<quote>
Given two angles and the side adjacent to them, the other angle is given.
</quote>
<lb/>
<s xml:id="echoid-s1447" xml:space="preserve">
Viète described four possible cases for this proposition; Harriot claims that he has missed some.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head180" xml:space="preserve" xml:lang="lat">
Vieta. resp. lib. 8. <lb/>
pag. 38. b.	<lb/>
Triangula ambigua
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 38v. <lb/>
Ambiguous triangles.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1449" xml:space="preserve">
Iisdem positis <lb/>
Quadrati etiam: <lb/>
duo casus omissi a Vieta
<lb/>[<emph style="it">tr:
The same things being supposed, the quadrants are also: <lb/>
Two cases missed by Viète.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1450" xml:space="preserve">
Inde <lb/>
Habendis angulus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> <lb/>
[???] duorum reliquorum <lb/>
angulorum.
<lb/>[<emph style="it">tr:
Having the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, hence the other two angles.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1451" xml:space="preserve">
Quadranti etiam: <lb/>
duo casus omissi a Vieta
<lb/>[<emph style="it">tr:
The quadrants are also: <lb/>
Two cases missed by Viète.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1452" xml:space="preserve">
Nota <lb/>
Obliangulorum sphæricorum <lb/>
duodecim sunt [???] <lb/>
quarum per perficuntu per <lb/>
syntomon et [???] <emph style="st">operatione</emph> <lb/>
[etc.]
<lb/>[<emph style="it">tr:
1. Two arcs each less tham a quadrant, and their complements, have equal differences. <lb/>
Let the two arcs be <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>, with difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. <lb/>
Their complements are <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>o</mi></mstyle></math>, also with difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f422v" o="422v" n="845"/>
<pb file="add_6782_f423" o="423" n="846"/>
<pb file="add_6782_f423v" o="423v" n="847"/>
<pb file="add_6782_f424" o="424" n="848"/>
<div xml:id="echoid-div270" type="page_commentary" level="2" n="270">
<p>
<s xml:id="echoid-s1453" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1453" 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 423.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head181" xml:space="preserve" xml:lang="lat">
Vieta. resp. lib. 8. <lb/>
pag. 43. b.
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 43v.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1455" xml:space="preserve">
Nota <lb/>
Ergo datis 2<emph style="super">bus</emph> lateribuset angulo complemento <lb/>
datur latus oppositum copendiose.
<lb/>[<emph style="it">tr:
Note <lb/>
Therefore given the two sides and the complement of the angle, the opposite side is given more briefly.
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s1456" xml:space="preserve">
See the papers of sines <lb/>
proportionall.
</s>
</p>
<pb file="add_6782_f424v" o="424v" n="849"/>
<pb file="add_6782_f425" o="425" n="850"/>
<pb file="add_6782_f425v" o="425v" n="851"/>
<pb file="add_6782_f426" o="426" n="852"/>
<div xml:id="echoid-div271" type="page_commentary" level="2" n="271">
<p>
<s xml:id="echoid-s1457" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1457" 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 423.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head182" xml:space="preserve" xml:lang="lat">
Vieta. resp. lib. 8. <lb/>
pag. 43. b. <lb/>
1. Datis tribus lateribis <lb/>
quæritur angulus A.
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 43v. <lb/>
Given three sides, there is sought angle A.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1459" xml:space="preserve">
Analogia <lb/>
Vide syntomon 2<emph style="super">o</emph>. <lb/>
Angulus quæsitis
<lb/>[<emph style="it">tr:
Ratio <lb/>
See syntomon 2. <lb/>
Angle sought.
</emph>]<lb/>
[<emph style="it">Note:
The second case of syntomon can be found on Add MS 6782, f. 421.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6782_f426v" o="426v" n="853"/>
<pb file="add_6782_f427" o="427" n="854"/>
<div xml:id="echoid-div272" type="page_commentary" level="2" n="272">
<p>
<s xml:id="echoid-s1460" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1460" 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 423.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head183" xml:space="preserve" xml:lang="lat">
Vieta. resp. lib. 8. <lb/>
pag. 43. b. <lb/>
1. Datis tribus lateribis <lb/>
quæritur angulus D.
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 43v. <lb/>
Given three sides, there is sought angle D.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1462" xml:space="preserve">
Analogia <lb/>
Vide syntomon 2<emph style="super">o</emph>. <lb/>
Angulus quæsitis
<lb/>[<emph style="it">tr:
Ratio <lb/>
See syntomon 2. <lb/>
Angle sought.
</emph>]<lb/>
[<emph style="it">Note:
The second case of syntomon can be found on Add MS 6782, f. 421.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6782_f427v" o="427v" n="855"/>
<pb file="add_6782_f428" o="428" n="856"/>
<div xml:id="echoid-div273" type="page_commentary" level="2" n="273">
<p>
<s xml:id="echoid-s1463" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1463" 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 423.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head184" xml:space="preserve" xml:lang="lat">
Vieta. resp. lib. 8. <lb/>
pag. 43. b. <lb/>
1. Datis tribus lateribis <lb/>
quæritur angulus B.
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 43v. <lb/>
Given three sides, there is sought angle B.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1465" xml:space="preserve">
Analogia <lb/>
numeratio in alia charta.
<lb/>[<emph style="it">tr:
Ratio <lb/>
Enumeration in the other sheet.
</emph>]<lb/>
[<emph style="it">Note:
The other sheet referred to here is probably Add MS 6782, f. 433.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6782_f428v" o="428v" n="857"/>
<pb file="add_6782_f429" o="429" n="858"/>
<div xml:id="echoid-div274" type="page_commentary" level="2" n="274">
<p>
<s xml:id="echoid-s1466" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1466" 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 423.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head185" xml:space="preserve" xml:lang="lat">
Vieta. resp. lib. 8. <lb/>
pag. 43. b.
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 43v.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1468" xml:space="preserve">
Menda in Vieta
<lb/>[<emph style="it">tr:
Wrong in Viète
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f429v" o="429v" n="859"/>
<pb file="add_6782_f430" o="430" n="860"/>
<div xml:id="echoid-div275" type="page_commentary" level="2" n="275">
<p>
<s xml:id="echoid-s1469" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1469" xml:space="preserve">
The reference to Fink is to <emph style="it">Geometriae rotundi libri XIIII</emph> (1583), page 364.
</s>
<lb/>
<s xml:id="echoid-s1470" 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-s1471" xml:space="preserve">
The reference to Viète is to the 'ALIUD' in Chapter XIX of
<emph style="it">Variorum resposorum liber VIII</emph>, Proposition 13.
See Add MS 6787, f. 223.
</s>
<lb/>
<quote xml:lang="lat">
13 Vt rectangulum quod sit sub sinu toto &amp; transsinuosa prima ad id quod sit
sub transsinuosa secunda &amp; transsinuosa tertia, ita quod sit sub sinu complementi secundæ
&amp; sinu complementi tertiæ ad id quod sit sub sinu toto &amp; sinu complemnti primæ.
</quote>
<lb/>
<quote>
As the product of the while sine and the secant of the first to
that of the secant of the second and the secant of the third,
so is that of the sine of the complement of the second and the sine of the complement of the third
to that of the whole sine and the sine of the complement of the first.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head186" xml:space="preserve" xml:lang="lat">
Finkius in Geomet. rotundi. <lb/>
lib. 14, 6. pag. 364. <lb/>
Regiom. lib.5.p.1.
<lb/>[<emph style="it">tr:
Fink in Geometria rotundi, Book XIV.6, page 364. <lb/>
Regiomontanus, Book V.1.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1473" xml:space="preserve">
Utilis propositio: <lb/>
ad indagendum angulum inclinationis <lb/>
circuli alicuius planetæ vel cometæ <lb/>
et ad alia.
<lb/>[<emph style="it">tr:
A useful proposition for delivering the angle of inclination of a circle of any planet or comet, and for other things.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1474" xml:space="preserve">
In duobus triangulis rectangulis <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>o</mi><mi>u</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>e</mi><mi>i</mi></mstyle></math>: <lb/>
Dantur <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>o</mi></mstyle></math>. latitudo plaentæ una <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>e</mi></mstyle></math>. latitudo altera <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>e</mi></mstyle></math>. differentia longitudinum
in duobus locis. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>i</mi></mstyle></math> datur ex consqequentia <lb/>
et est arcus circuli <lb/>
planetæ. <lb/>
Quæritur angulus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
In two right-angled triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>o</mi><mi>u</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>e</mi><mi>i</mi></mstyle></math>, there are given: <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>o</mi></mstyle></math>, the latitude of one planet <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>e</mi></mstyle></math>, the latitude of the other <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>e</mi></mstyle></math>, the difference in longitude of the two locations.
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>i</mi></mstyle></math> is consequently given, and is the arc of a circle of a planet. <lb/>
There is sought angle <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-s1475" xml:space="preserve">
Inde per Finkium <lb/>
<lb/>[...]<lb/> <lb/>
Quas Analogias deduxit ex superioribus ita: <lb/>
<lb/>[...]<lb/> <lb/>
Sed ita nullum compendium oritur, igitur inutilis commutatio.
<lb/>[<emph style="it">tr:
Thus by Fink. <lb/>
<lb/>[...]<lb/> <lb/>
Whcih ratios one deduces from the above, thus: <lb/>
<lb/>[...]<lb/> <lb/>
But in this way nothing shorter arises, therefore the change is not useful.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1476" xml:space="preserve">
Utile compendium ita fit <lb/>
<lb/>[...]<lb/> <lb/>
Latitudo, inventa <lb/>
per syntomon.
<lb/>[<emph style="it">tr:
It may usefully be done more briefly thus: <lb/>
<lb/>[...]<lb/> <lb/>
The latitude is found by syntomon.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1477" xml:space="preserve">
vel per 13p. vieta lib. 8. resp. pag. 37. <lb/>
<lb/>[...]<lb/> <lb/>
Hoc est:<lb/>
latitudo inventa <lb/>
+ proportione <lb/>
Hæc mutatio ergo inutilis: <lb/>
vel hæc melior quam <lb/>
illa Finkij vel originis.
<lb/>[<emph style="it">tr:
or by Proposition 13 of Viète, Responsorum liber VIII, page 37, <lb/>
<lb/>[...]<lb/> <lb/>
That is:<lb/>
the latitude found, and the proportion <lb/>
This change is therefore not useful; or this is better than that of Fink or the original.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f430v" o="430v" n="861"/>
<pb file="add_6782_f431" o="431" n="862"/>
<pb file="add_6782_f431v" o="431v" n="863"/>
<pb file="add_6782_f432" o="432" n="864"/>
<div xml:id="echoid-div276" type="page_commentary" level="2" n="276">
<p>
<s xml:id="echoid-s1478" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1478" 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>
a diagram realting to these figures is to be found on page 426.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head187" xml:space="preserve" xml:lang="lat">
Vieta. pag. 45. <lb/>
resp. lib. 8.
<lb/>[<emph style="it">tr:
Viète, page 45, Responsorum liber VIII, page 43.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1480" xml:space="preserve">
Anguli obliquanguli <lb/>
trianguli sphæricæ.
<lb/>[<emph style="it">tr:
Oblique-angled spherical triangles.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f432v" o="432v" n="865"/>
<pb file="add_6782_f433" o="433" n="866"/>
<div xml:id="echoid-div277" type="page_commentary" level="2" n="277">
<p>
<s xml:id="echoid-s1481" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1481" 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 triangles referred to here are to be found on pages 422 and 423.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head188" xml:space="preserve" xml:lang="lat">
Vieta. rep. lib. 8 <lb/>
pag. 43. <lb/>
Anguli, rectanguli trianguli sphæricæ
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 43. Angles, in right-angled spherical triangles.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1483" xml:space="preserve">
omnes combinationes faciunt trianguli. <lb/>
sunt quatuor tantum quia unus est 90.
<lb/>[<emph style="it">tr:
all combinatins make triangles; there are four such because one is 90.
</emph>]<lb/>
</s>
</p>
<head xml:id="echoid-head189" xml:space="preserve" xml:lang="lat">
pag. 44. Anguli obliquianguli
<lb/>[<emph style="it">tr:
page 44. Oblique angles
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1484" xml:space="preserve">
Nullam earum combinationium <lb/>
faciunt trainguli. <lb/>
Accipi igitur complementum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> <lb/>
ut facit Vieta. ita:
<lb/>[<emph style="it">tr:
None of these combinations makes a triangle. <lb/>
Therefore accept the complememnt of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> as Viète does, thus:
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f433v" o="433v" n="867"/>
<pb file="add_6782_f434" o="434" n="868"/>
<pb file="add_6782_f434v" o="434v" n="869"/>
<pb file="add_6782_f435" o="435" n="870"/>
<div xml:id="echoid-div278" type="page_commentary" level="2" n="278">
<p>
<s xml:id="echoid-s1485" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1485" xml:space="preserve">
The text referred to here is Johan Philip Lansberg,
<emph style="it">Triangulorum geometriae libri quatuor</emph> (1591).
Page 201 contains Lansberg's rule for finding a side of a spherical triangles, given its angles.
See Add MS 6787, f. 197.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head190" xml:space="preserve">
Erallage pleuroniniche
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1487" xml:space="preserve">
Lansberg. pag. 201. Demonstratio originis falsa est: <lb/>
Regularum aliquando
<lb/>[<emph style="it">tr:
Lansberg, page 201. The original demonstration is false for some rules.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f435v" o="435v" n="871"/>
<pb file="add_6782_f436" o="436" n="872"/>
<pb file="add_6782_f436v" o="436v" n="873"/>
<pb file="add_6782_f437" o="437" n="874"/>
<pb file="add_6782_f437v" o="437v" n="875"/>
<pb file="add_6782_f438" o="438" n="876"/>
<div xml:id="echoid-div279" type="page_commentary" level="2" n="279">
<p>
<s xml:id="echoid-s1488" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1488" xml:space="preserve">
A continuation of Harriot's work on the 'Dati sexti', from Chapter XIX of
Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593).
Here he examines Proposition VI.
The word 'parapompe' was originally used by Viète to describe each proposition. <lb/>
</s>
<lb/>
<quote xml:lang="lat">
VII. <lb/>
Data summa vel differentia duarum perpheriarum, quarum prosinus datam habeant rationem,
dantur singulæ <lb/>
1 Enimvero si utraque peripheria proponatur minor quadrante, vel utraque major. <lb/>
Erit, <lb/>
Vt adgregatum similium prosinuum ad differentiam eorundem,
ita sinus summæ peripheriarum ad sinuum differentiæ,
Vel ita transsinuosa complementi differentiæ ad transsinuosam complementi summæ. <lb/>
2 Quod si una e peripheriis proponatur minor quadrante, altera maior, <lb/>
Erit, <lb/>
Vt adgregatum prosinuum ad differentiam eorundem,
ita sinus differentiæ peripheriarum ad sinum adgregati,
Vel ita transsinuosa complementi summæ ad transsinuosam complementi differentiæ.
</quote>
<lb/>
<quote>
VII. Given the sum or difference of two arcs, whose tangents are in a given ratio, each is given individually. <lb/>
1. If both given arcs are less than a quadrant, or both greater,
then as the sum of the tangents is to their difference,
so is the sine of the sum of the arcs to the sine of their difference. <lb/>
Or as the secant of the complement of the difference to the secant of the complement of the sum. <lb/>
2. But if one of the given arcs is less than a quadrant, the other greater,
then as the sum of the tangents is to their difference,
so is the sine of the difference of the arcs to the sine of the sum. <lb/>
Or as the secant of the complement of the sum to the secant of the complement of the difference.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head191" xml:space="preserve" xml:lang="lat">
Vieta. 37.b. lib. 8. resp. <lb/>
<foreign xml:lang="gre">parapompe</foreign> <lb/>
Dati Septimi. <lb/>
Data summa vel differentia duarum peripheriarum, Â <lb/>
et ratione O.
<lb/>[<emph style="it">tr:
Viète, page 37v, Responsorum liber VIII. <lb/>
Parapompe <lb/>
Seventh proposition. <lb/>
Given the sum or difference of two arcs and the ratio of their tangents.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1490" xml:space="preserve">
1. <lb/>
utraque minor <lb/>
quadrante
<lb/>[<emph style="it">tr:
1. both less than a quadrant
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1491" xml:space="preserve">
Interpetatio <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, una peripheria, minor quadrante. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, altera peripheria minor quadrante. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi><mo>=</mo><mi>b</mi><mi>a</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>c</mi></mstyle></math>, differentia peripheriæ. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi></mstyle></math>, tangens <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math>, tangens <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>f</mi></mstyle></math>, differentia tangentium. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi></mstyle></math>, sinus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>l</mi></mstyle></math>, sinus differentiæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Interpetation <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, one arc, less than a quadrant. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, the other arc, less than a quadrant. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi><mo>=</mo><mi>b</mi><mi>a</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>c</mi></mstyle></math>, the difference of the arcs. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi></mstyle></math>, tangent to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math>, tangento <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>f</mi></mstyle></math>, the difference of the tangents. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi></mstyle></math>, sine of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>l</mi></mstyle></math>, sine of the difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1492" xml:space="preserve">
2. <lb/>
una maior
<lb/>[<emph style="it">tr:
2. one greater
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1493" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> peripheria maior quadrante. <lb/>
cætera ut supra
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> is an arc greater than a quadrant; <lb/>
the rest is as above.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f438v" o="438v" n="877"/>
<pb file="add_6782_f439" o="439" n="878"/>
<div xml:id="echoid-div280" type="page_commentary" level="2" n="280">
<p>
<s xml:id="echoid-s1494" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1494" xml:space="preserve">
A continuation of Harriot's work on the 'Dati sexti', from Chapter XIX of
Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593).
Here he examines Proposition VI.
</s>
<lb/>
<quote xml:lang="lat">
VI. <lb/>
Data summa vel differentia duarum perpheriarum, quarum sinus datam habeant rationem,
dantur singulares peripheriæ. <lb/>
1 Enimvero si utraque peripheria proponitur minor quadrante, vel utraque major. <lb/>
Erit, <lb/>
Vt adgregatum similium sinuum ad differentiam eorundem,
ita prosinus dimidiæ summæ peripheriarum ad prosinum dimidiæ differentiæ earundem,
Vel ita prosinus complementi dimidiæ differentiæ peripheriarum ad prosinum complementi dimidiæ summæ. <lb/>
2 Quod si una e peripheriis proponatur minor quadrante, altera maior, <lb/>
Erit, <lb/>
Vt adgregatum sinuum ad differentiam eorundem,
ita prosinus dimidiæ differentiæ peripheriarum ad prosinum dimidiæ summæ,
Vel ita prosinus complementi dimidiæ summæ ad prosinum complementi dimidiæ differentiæ.
</quote>
<lb/>
<quote>
VI. Given the sum or difference of two arcs, whose sines are in a given ratio, each arc is given individually. <lb/>
1. If both given arcs are less than a quadrant of the circle, or both greater,
then as the sum of those sines is to their difference,
so is the tangent of half the sum of the arcs to the tangent of half their difference. <lb/>
Or as the tangent of the complement of half the difference of the arcs to
the tangent of the complement of half the sum. <lb/>
2. But if one of the given arcs is less than a quadrant, the other greater,
then as the sum of the sines is to their difference,
so is the tangent of half the difference of the arcs to the tangent of half their sum. <lb/>
Or as the tangent of the complement of half the sum to the tangent of the complement of half the difference.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head192" xml:space="preserve" xml:lang="lat">
Vieta. lib. 8. resp. <lb/>
pag. 37. <lb/>
VI. <lb/>
Data summa vel differentia <lb/>
et ratione <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Î¥</mo></mstyle></math>.
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 37, VI. <lb/>
Given the sum or difference and the ratio of their sines.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1496" xml:space="preserve">
Interpetatio <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, una peripheria minor quadrante. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, altera peripheria minor quadrante. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>g</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi></mstyle></math> ratio sinuum <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>=</mo><mi>m</mi><mi>c</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>m</mi></mstyle></math> differentia inter <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>b</mi><mi>c</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>h</mi></mstyle></math> differentia inter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>g</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Interpetation <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> one arc, less than a quadrant <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> the other arc, less than a quadrant <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>g</mi><mo>:</mo><mi>g</mi><mi>c</mi></mstyle></math> ratio of the sines <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>=</mo><mi>m</mi><mi>c</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>m</mi></mstyle></math> 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>b</mi><mi>c</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>h</mi></mstyle></math> difference between <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>g</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1497" xml:space="preserve">
Hic <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> est maior quadrante. non tamen variat casum. <lb/>
Menda igitur in Vieta
<lb/>[<emph style="it">tr:
Here <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> is greater than a quadrant, nevertheless, the case does not change. <lb/>
Therefore wrong in Viète.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f439v" o="439v" n="879"/>
<pb file="add_6782_f440" o="440" n="880"/>
<div xml:id="echoid-div281" type="page_commentary" level="2" n="281">
<p>
<s xml:id="echoid-s1498" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1498" xml:space="preserve">
A continuation of Harriot's work on the 'Dati sexti', from Chapter XIX of
Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593).
Here he examines Propositions V.1 and V.2.
</s>
<lb/>
<quote xml:lang="lat">
V. <lb/>
Data differentia duarum perpheriarum, quarum sinus datam habeant rationem, dantur singulæ <lb/>
1 Enimvero si differentia sit maior quadrante circuli. <lb/>
Erit, <lb/>
Vt sinus componentium primæ ad sinum secundæ,
ita transsinuosa complementi differentiæ ad prosinum complementi primæ minus prosinu complementi differentiæ. <lb/>
Cum autem prima sumetur maior quadrante, secunda sumetur minor, &amp; contra. <lb/>
2 Et si differentia minor quadrante circuli, differentes autem peripheriæ diversæ sint speciei, <lb/>
Erit, <lb/>
Vt sinus primæ ad sinum secundæ,
ita transsinuosa complementi differentiæ ad prosinum complementi differentiæ, plus prosinu complementi primæ. <lb/>
Cum autem prima sumetur maior quadrante, secunda sumetur minor, &amp; contra.
</quote>
<lb/>
<quote>
V. Given the difference of two arcs, whose sines are in a given ratio, each is given individually. <lb/>
1. If the difference is greater than a quadrant of the circle,
then as the sine of the first component is to the sine of the second,
so is the secant of the complement of the difference to the tangent of the complement of the first
minus the tangent of the complement of the difference. <lb/>
Moreover, when the first it taken greater than a quadrant, the second is taken less, and conversely. <lb/>
2. And if the difference is greater than a quadrant of the circle, but the different arcs have different signs,
then as the sine of the first component is to the sine of the second,
so is the secant of the complement of the difference to the tangent of the complement of the difference
plus the tangent of the complement of the first.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head193" xml:space="preserve" xml:lang="lat">
Vieta. lib. 8. resp. <lb/>
pag. 37. <lb/>
V. <lb/>
Data differentia duarum perpheriarum <lb/>
et ratione <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Î¥</mo></mstyle></math>.
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 37, V. <lb/>
Given the difference of two arcs and the ratio of their sines.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1500" xml:space="preserve">
1. <lb/>
differentia <lb/>
maior quad.
<lb/>[<emph style="it">tr:
1. the difference greater than a quadrant
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1501" xml:space="preserve">
Interpetatio <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> una peripheria. cui æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> altera <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi></mstyle></math> differentia <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>h</mi></mstyle></math> angulus rectus <lb/>
<lb/>[...]<lb/>
<lb/>[<emph style="it">tr:
Interpetation <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> one arc, to which <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi></mstyle></math> is equal <lb/>
<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>b</mi><mi>g</mi></mstyle></math> the difference
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>h</mi></mstyle></math>, a right angle <lb/>
<lb/>[...]<lb/>
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1502" xml:space="preserve">
Ut minor terminus sit primum proportionalium. <lb/>
Fiat angulus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>e</mi><mi>r</mi></mstyle></math>, æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>k</mi></mstyle></math> angulo, qui est angulus complementi differentiæ. <lb/>
Tum triangula <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>r</mi><mi>g</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi><mi>k</mi><mi>e</mi></mstyle></math> sunt æquiangula. nam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi><mi>k</mi><mi>e</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>r</mi><mi>g</mi></mstyle></math> sunt anguli <lb/>
residui æqualia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>r</mi><mi>e</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>k</mi><mi>e</mi></mstyle></math>. et anguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>e</mi><mi>l</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>g</mi><mi>e</mi></mstyle></math> sunt æquales ab paralleles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>q</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>l</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
If the smaller term is the first proportional. <lb/>
Construct anlge <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>e</mi><mi>r</mi></mstyle></math> equal to angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>k</mi></mstyle></math>, which is the angle of the complement of the difference. <lb/>
Then triangles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>r</mi><mi>g</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi><mi>k</mi><mi>e</mi></mstyle></math> are equiangular, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi><mi>k</mi><mi>e</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>r</mi><mi>g</mi></mstyle></math> are residual angles from
the equal angles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>r</mi><mi>e</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>k</mi><mi>e</mi></mstyle></math>; and angles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>e</mi><mi>l</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>g</mi><mi>e</mi></mstyle></math> are equals by the parallels <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>q</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>l</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1503" xml:space="preserve">
2. <lb/>
differentia <lb/>
minor quad: <lb/>
peripheria <lb/>
una minor, <lb/>
altera maior <lb/>
quadrante.
<lb/>[<emph style="it">tr:
2. the difference less than a quadrant; one arc less tha, the other greater than a quadrant.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1504" xml:space="preserve">
Superiora verba et litteræ <lb/>
deservierunt etiam hinc diagram-<lb/>
mati; et concludant:
<lb/>[<emph style="it">tr:
The above words and letters serve also for this diagram; and end with:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1505" xml:space="preserve">
Ut minor terminus sit primus proportionalium. <lb/>
Hic anguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>r</mi><mi>g</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>k</mi><mi>l</mi></mstyle></math>, sunt complemmentat <lb/>
æqualium angulum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>e</mi><mi>n</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>e</mi><mi>h</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
As the lesser term is the first proportional. <lb/>
Here angles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>r</mi><mi>g</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>k</mi><mi>l</mi></mstyle></math> are complements of equal angles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>e</mi><mi>n</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>e</mi><mi>h</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f440v" o="440v" n="881"/>
<pb file="add_6782_f441" o="441" n="882"/>
<div xml:id="echoid-div282" type="page_commentary" level="2" n="282">
<p>
<s xml:id="echoid-s1506" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1506" xml:space="preserve">
A continuation of Harriot's work on the 'Dati sexti', from Chapter XIX of
Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593).
Here he examines Proposition V.3.
</s>
<lb/>
<quote xml:lang="lat">
V. <lb/>
Data differentia duarum perpheriarum, quarum sinus datam habeant rationem, dantur singulæ <lb/>
<lb/>[...]<lb/> <lb/>
3 Et si denique differentia sit minor quadrante,
utraque vero differentium vel quadrante minor vel utraque quadrante maior,
ac prima quidem intelligatur ea cui debetur sinus major, secunda cui minor, <lb/>
Erit, <lb/>
Vt sinus primæ ad sinum secundæ,
ita transsinuosa complementi differentiæ ad prosinum complementi differentiæ minus prosinu complementi primæ. <lb/>
Et, <lb/>
Vt sinus primæ ad sinum secundæ,
ita transsinuosa complementi differentiæ ad prosinum complementi differentiæ plus prosinu complementi primæ.
</quote>
<lb/>
<quote>
V. Given the difference of two arcs, whose sines are in a given ratio, each is given individually. <lb/>
<lb/>[...]<lb/> <lb/>
3. And if finally the difference is less than a quadrant,
then as the sine of the first component is to the sine of the second,
so is the secant of the complement of the difference to the tangent of the complement of the difference
minus the tangent of the complement of the first.
And as the sine of the first is to the sine of the second,
so is the secant of the complement of the difference to the tangent of the complement of the difference
plus the tangent of the complement of the first. <lb/>
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head194" xml:space="preserve" xml:lang="lat">
Vieta. lib. 8. resp. pag. 37. <lb/>
V. <lb/>
Data differentia duarum perpheriarum <lb/>
et ratione <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Î¥</mo></mstyle></math>.
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 37, V. <lb/>
Given the difference of two arcs and the ratio of their sines.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1508" xml:space="preserve">
3. <lb/>
differentia <lb/>
minor quad. <lb/>
et <lb/>
utraque <lb/>
peripheriæ.
<lb/>[<emph style="it">tr:
3. the difference greater than a quadrant, and both the arcs.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1509" xml:space="preserve">
Verba <emph style="super">et litteræ</emph> superiores <lb/>
diagrammatis <lb/>
2, et 1, deservierunt <lb/>
etiam huic.
<lb/>[<emph style="it">tr:
The words and letters for the above diagrams, 2 and 1, serve also for this,
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1510" xml:space="preserve">
Ut minor terminus sit primum proportinalium.
<lb/>[<emph style="it">tr:
If the smaller term is the first proportional.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f441v" o="441v" n="883"/>
<pb file="add_6782_f442" o="442" n="884"/>
<div xml:id="echoid-div283" type="page_commentary" level="2" n="283">
<p>
<s xml:id="echoid-s1511" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1511" xml:space="preserve">
A continuation of Harriot's work on the 'Dati sexti', from Chapter XIX of
Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593).
Here he examines Proposition IV.
</s>
<lb/>
<quote xml:lang="lat">
IV. <lb/>
Data peripheria composita e duabus peripheriis, quarum sinus datam habeant rationem, dantur singulæ. <lb/>
1 Enimvero si composita minor est circuli quadrante. <lb/>
Erit, <lb/>
Vt sinus componentium primæ ad sinum secundæ,
ita transsinuosa complementi compositæ ad prosinum complementi primæ minus prosinu complementi compositæ. <lb/>
2 Et si composita maior est quadrante, utraque vero componentium minor quadrante. <lb/>
Erit, <lb/>
Vt sinus primæ ad sinum secundæ,
ita s complementi compositæ ad prosinum complementi compositæ plus prosinu complementi primæ. <lb/>
3 Et si denique componentium peripheriarum primæ sit minor quadrante, secunda maior, <lb/>
Erit, <lb/>
Vt sinus primæ ad sinum secundæ,
ita transsinuosa complementi compositæ ad prosinum complementi compositæ minus prosinu complementi primæ. <lb/>
Et, <lb/>
Vt sinus primæ ad sinum secundæ,
ita transsinuosa complementi compositæ ad prosinum complementi compositæ plus prosinu secundæ. <lb/>
</quote>
<lb/>
<quote>
IV. Given the sum of two arcs, whose sines are in a given ratio, each is given individually. <lb/>
1. If the sum is less than a quadrant of the circle, then as the sine of the first component is to the sine of the second,
so is the secant of the complement of the sum to the tangent of the complement of the first
minus the tangent of the complement of the sum. <lb/>
2. And if the sum is greater than a quadrant, but both components are less than a quadrant,
then as the sine of the first component is to the sine of the second,
so is the secant of the complement of the sum to the tangent of the complement of the sum
plus the tangent of the complement of the first. <lb/>
3. And if finally the first component of the sum is less than a quadrant, the second greater,
then as the sine of the first component is to the sine of the second,
so is the secant of the complement of the sum to the tangent of the complement of the sum
minus the tangent of the complement of the first.
And as the sine of the first is to the sine of the second,
so is the secant of the complement of the sum to the tangent of the complement of the sum
plus the tangent of the second.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head195" xml:space="preserve" xml:lang="lat">
Vieta. lib. 8. resp. <lb/>
pag. 38. b. <lb/>
IIII. <lb/>
Data peripheria composita <lb/>
et ratione <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Î¥</mo></mstyle></math>.
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 38v, IV. <lb/>
Given a sum of arcs and the ratio of their sines, 2.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1513" xml:space="preserve">
1. <lb/>
composita minor <lb/>
quadrante
<lb/>[<emph style="it">tr:
1. the sum less than a quadrant
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1514" xml:space="preserve">
Interpetatio <lb/>
<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>a</mi><mi>d</mi></mstyle></math>,<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math>: ratio sinuum <lb/>
<lb/>[...]<lb/>
<lb/>[<emph style="it">tr:
Interpetation <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> one arc <lb/>
<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>a</mi><mi>d</mi><mo>:</mo><mi>d</mi><mi>c</mi></mstyle></math>, the ratio of sines <lb/>
<lb/>[...]<lb/>
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1515" xml:space="preserve">
2. <lb/>
composita maior; <lb/>
utraque minor
<lb/>[<emph style="it">tr:
2. the sum greater; both [arcs] less
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1516" xml:space="preserve">
3. <lb/>
composita <lb/>
maior: <lb/>
una minor, <lb/>
altera maior
<lb/>[<emph style="it">tr:
3. the sum greater; one [arc] less, the other greater
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1517" xml:space="preserve">
Inde cum 2. <lb/>
Menda in Vieta
<lb/>[<emph style="it">tr:
Hence like 2, wrong in Viète.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1518" xml:space="preserve">
Aliter pro 3. Ut maior terminus sit primus proportionalium. <lb/>
<lb/>[<emph style="it">tr:
Another way for 3, when the greater term is the first proportional.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f442v" o="442v" n="885"/>
<pb file="add_6782_f443" o="443" n="886"/>
<div xml:id="echoid-div284" type="page_commentary" level="2" n="284">
<p>
<s xml:id="echoid-s1519" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1519" xml:space="preserve">
A continuation of Harriot's work on the 'Dati sexti', from Chapter XIX of
Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593).
Here he examines Proposition III.2.
</s>
<lb/>
<quote xml:lang="lat">
III. <lb/>
Data summa vel differentia duarum peripheriarum, quarum transsinuosae datam habeant rationem, dantur singulæ. <lb/>
<lb/>[...]<lb/> <lb/>
2 Quod si une e peripheriis proponitur minor quadrante, altera maior <lb/>
Erit, <lb/>
Vt adgregatum similium transsinuousuarum ad differentiam earundem,
ita prosinus dimidia differentiæ peripheriarum ad prosinum complementi dimidæ summæ,
Et ita prosinus dimidiæ summæ ad prosinum complementi dimidiæ differentiæ.
</quote>
<lb/>
<quote>
III. Given the sum or difference of two arcs, whose secants are in a given ratio, each is given individually. <lb/>
2. But if one of the arcs is less than a quadrant, the ohter greater, then as the sum of the similar secants is
to their difference, so is the tangent of half the difference of the arcs
to the tangent of the complement of half the sum.
And so is the tangent of half the sum to the tangent of the complement of half the difference.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head196" xml:space="preserve" xml:lang="lat">
Vieta. lib. 8. resp. <lb/>
pag. 38. b. <lb/>
III. <lb/>
Data summa vel differentia <lb/>
et ratione <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math>. 2.
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 38v, III. <lb/>
Given the sum or difference of two arcs and the ratio of their secants, 2.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1521" xml:space="preserve">
2. <lb/>
peripheria <lb/>
una minor <lb/>
quadrante; <lb/>
altera maior.
<lb/>[<emph style="it">tr:
2. one arc is less than a quadrant, the other greater
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1522" xml:space="preserve">
Interpetatio <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> una peripheria, minor quadrante <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> altera, maior quadrante <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>i</mi><mo>-</mo><mi>a</mi><mi>b</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>c</mi></mstyle></math>, 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>b</mi><mi>c</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>c</mi></mstyle></math>, dimidia differentia <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>g</mi></mstyle></math>, eius tangens <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>g</mi><mo>=</mo><mi>k</mi><mi>g</mi></mstyle></math>, et parallelæ <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi></mstyle></math> secans peripheriæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi></mstyle></math>, secans peripheriæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>t</mi><mo>=</mo><mi>e</mi><mi>d</mi><mo>=</mo><mi>e</mi><mi>f</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi><mi>h</mi></mstyle></math>, differentia secantium <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi></mstyle></math> <lb/>
fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>y</mi><mo>=</mo><mi>e</mi><mi>t</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>i</mi><mi>k</mi></mstyle></math> dimidia summa compositæ <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>m</mi></mstyle></math>, complementum dimidiæ summæ <lb/>
<lb/>[...]<lb/>
<lb/>[<emph style="it">tr:
Interpetation <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> one arc, less than a quadrant <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> the other, greater than a quadrant <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>i</mi><mo>-</mo><mi>a</mi><mi>b</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>c</mi></mstyle></math>, 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>b</mi><mi>c</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>c</mi></mstyle></math>, half the difference <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>g</mi></mstyle></math>, its tangent <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>g</mi><mo>=</mo><mi>k</mi><mi>g</mi></mstyle></math>, and parallels <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi></mstyle></math> secant of the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi></mstyle></math>, secant of the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>t</mi><mo>=</mo><mi>e</mi><mi>d</mi><mo>=</mo><mi>e</mi><mi>f</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi><mi>h</mi></mstyle></math>, difference of the secants <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi></mstyle></math> <lb/>
fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>y</mi><mo>=</mo><mi>e</mi><mi>t</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>i</mi><mi>k</mi></mstyle></math> half the sum of the composite arc <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>m</mi></mstyle></math>, complement of half the sum <lb/>
<lb/>[...]<lb/>
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f443v" o="443v" n="887"/>
<pb file="add_6782_f444" o="444" n="888"/>
<div xml:id="echoid-div285" type="page_commentary" level="2" n="285">
<p>
<s xml:id="echoid-s1523" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1523" xml:space="preserve">
A continuation of Harriot's work on the 'Dati sexti', from Chapter XIX of
Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593).
Here he examines Proposition III.1.
The word 'parapompe' was originally used by Viète to describe each proposition. <lb/>
The page number 36v given in the top left hand corner is incorrect; it should be 38v.
</s>
<lb/>
<quote xml:lang="lat">
III. <lb/>
Data summa vel differentia duarum peripheriarum, quarum transsinuosae datam habeant rationem, dantur singulæ. <lb/>
1 Enimvero si utraque peripheria proponatur minor quadrante vel utraque major. <lb/>
Erit, <lb/>
Vt adgregatum similium transsinuousuarum ad differentiam earundem,
ita prosinus complementi dimidia summæ peripheriæ ad prosinum dimidæ differentiæ,
Et ita prosinus complementi dimidiæ differentiæ ad prosinum dimidiæ summæ.
</quote>
<lb/>
<quote>
III. Given the sum or difference of two arcs, whose secants are in a given ratio, each is given individually. <lb/>
1. If both the given arcs are less than a quadrant or both greater, then as the sum of the similar secants is
to their difference, so is the tangent of the complement of half the sum of the arcs
to the tangent of half the difference.
And so is the tangent of the complement of half the difference to the tangent of half the sum.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head197" xml:space="preserve" xml:lang="lat">
Vieta. lib. 8. resp. <lb/>
pag. 36. b. <lb/>
III. <lb/>
et pag. 38. <lb/>
1. <lb/>
data peripheria <lb/>
composita. <lb/>
et ratione <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Î¨</mo></mstyle></math>. <lb/>
<foreign xml:lang="gre">parapompe</foreign>. 3. Data summa vel differentia duarum peripheriarum <lb/>
et ratione <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math>. 1.
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 36v, III. <lb/>
and page 38, 1, given the sum of arcs and the ratio of their secants <lb/>
Parapompe III: Given the sum or difference of two arcs and the ratio of their secants.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1525" xml:space="preserve">
I.1. <lb/>
peripheria utraque <lb/>
minor quadrante
<lb/>[<emph style="it">tr:
I.1.either arc is less than a quadrant
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1526" xml:space="preserve">
Arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. Tangens <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi></mstyle></math>. <lb/>
Arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. Tangens <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>=</mo><mi>b</mi><mi>i</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>c</mi><mo>,</mo><mi>d</mi><mi>i</mi><mi>f</mi><mi>f</mi><mi>e</mi><mi>r</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>a</mi><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi></mstyle></math> ab <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>t</mi></mstyle></math> bc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>.</mo></mstyle></math><lb/>
Aggregatum tangentium <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi><mo>+</mo><mi>e</mi><mi>h</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>h</mi></mstyle></math>, est differentia tangentium <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>k</mi></mstyle></math> est dimidium arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>c</mi></mstyle></math>. <lb/>
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>r</mi></mstyle></math> est dimidium totius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>s</mi></mstyle></math> est complementum arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>r</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>n</mi></mstyle></math> est tangens complementi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>k</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>l</mi></mstyle></math> est tangens arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>k</mi></mstyle></math>, secans <lb/>
dimidij differentiæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>c</mi></mstyle></math>. <lb/>
Lineæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>n</mi></mstyle></math> sit parallellæ
<lb/>[<emph style="it">tr:
Arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, tangent <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi></mstyle></math>. <lb/>
Arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, tangens <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>=</mo><mi>b</mi><mi>i</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>c</mi><mo>,</mo><mi>t</mi><mi>h</mi><mi>e</mi><mi>d</mi><mi>i</mi><mi>f</mi><mi>f</mi><mi>e</mi><mi>r</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>b</mi><mi>e</mi><mi>t</mi><mi>w</mi><mi>e</mi><mi>e</mi><mi>n</mi></mstyle></math> ab <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>n</mi><mi>d</mi></mstyle></math> bc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>.</mo></mstyle></math><lb/>
Sum of the tangents <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi><mo>+</mo><mi>e</mi><mi>h</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>h</mi></mstyle></math> is the difference between the tangents <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>k</mi></mstyle></math> is half the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>c</mi></mstyle></math>. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>r</mi></mstyle></math> is half the total <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>s</mi></mstyle></math> is the complement of the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>r</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>n</mi></mstyle></math> is the tangent of the complement <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>k</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>l</mi></mstyle></math> is the tangent of the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>k</mi></mstyle></math>, cutting half the difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>c</mi></mstyle></math>. <lb/>
The lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>n</mi></mstyle></math> are parallel.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1527" xml:space="preserve">
<foreign xml:lang="gre">parapompe</foreign> pro pag. 38. Data peripheria <lb/>
Triangles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>d</mi><mi>e</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>x</mi><mi>u</mi></mstyle></math> are equiangular <lb/>
For angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>u</mi><mi>x</mi></mstyle></math> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>d</mi></mstyle></math>, because either is the complement of angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>e</mi><mi>u</mi></mstyle></math>,
and it is obvious that angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi><mi>h</mi></mstyle></math> is equal to angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>x</mi><mi>e</mi></mstyle></math>;
therefore a third of angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>u</mi><mi>x</mi></mstyle></math> is equal to angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi><mi>d</mi></mstyle></math>. <lb/>
<lb/>[...]<lb/> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>a</mi></mstyle></math>, is the tangent of the conplement of the sum. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>x</mi></mstyle></math>, is the tangent of the second.
<lb/>[<emph style="it">tr:
<foreign xml:lang="gre">parapompe</foreign> for page 38. Given the sum of the arcs. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>=</mo><mi>b</mi><mi>i</mi></mstyle></math> <lb/>
Triangula <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>d</mi><mi>e</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>x</mi><mi>u</mi></mstyle></math> sunt æquiangula <lb/>
Nam angulus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>u</mi><mi>x</mi></mstyle></math> est æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>d</mi></mstyle></math> quia <lb/>
uterque est complementum anguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>e</mi><mi>u</mi></mstyle></math> <lb/>
Et manifestum est angulus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi><mi>h</mi></mstyle></math> esse <lb/>
æqualem angulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>x</mi><mi>e</mi></mstyle></math>; Ergo tertius <lb/>
angulus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>u</mi><mi>x</mi></mstyle></math> æqualis est angulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi><mi>d</mi></mstyle></math>. <lb/>
<lb/>[...]<lb/> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>a</mi></mstyle></math>, est prosinus complementi compositæ <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>x</mi></mstyle></math>, est prosinus secundæ.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f444v" o="444v" n="889"/>
<pb file="add_6782_f445" o="445" n="890"/>
<div xml:id="echoid-div286" type="page_commentary" level="2" n="286">
<p>
<s xml:id="echoid-s1528" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1528" xml:space="preserve">
At the end of Chapter XIX of <emph style="it">Variorum responsorum liber VIII</emph> (1593),
under the heading 'DATI SEXTI', Viète listed six propositions for finding either of two arcs
given the sums of differences of the arcs and the ratio of their secants.
In the 1646 edition of Viète's <emph style="it">Opera mathematica</emph>
the six propositions are to be found on page 413.
</s>
<lb/>
<s xml:id="echoid-s1529" xml:space="preserve">
On this page Harriot examines Proposition I, part 2.
</s>
<lb/>
<quote xml:lang="lat">
I. <lb/>
Data peripheria composita e duabus peripheriis, quarum transsinuosae datam habeant rationem, dantur singulæ. <lb/>
<lb/>[...]<lb/> <lb/>
2 Et si composita major est quadrante circuli, utraque vero componentium minor quadrante, <lb/>
Erit, <lb/>
Vt transsinuosa primæ ad transsinuousa secundæ, ita transsinuosa complementi composita ad prosinum secunda
minus prosinus complementa compositæ.
</quote>
<lb/>
<quote>
I.Given an arc composed of two others, whose secants are in a given ratio, each is known individually. <lb/>
<lb/>[...]<lb/> <lb/>
2. And if the sum is greater than a quarter of a circle, but each component is less than a quarter, then
as the secant of the first is to the secant of the second, so is the secant of the complement of the sum
to the tangent of the second minus the tangent of the complement of the sum.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head198" xml:space="preserve" xml:lang="lat">
Vieta. lib. 8. resp. <lb/>
pag. 38. <lb/>
Data peripheria composita duarum perpheriarum, <lb/>
et ratione <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 38. <lb/>
Given an arc composed of two arcs, and the ratio of their secants.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1531" xml:space="preserve">
composita <lb/>
maior <lb/>
utrusque minor
<lb/>[<emph style="it">tr:
the sum is greater or lesser than
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1532" xml:space="preserve">
triangula <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>d</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi><mi>g</mi></mstyle></math> sunt equiangula. nam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>e</mi><mi>g</mi></mstyle></math> æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi><mi>d</mi></mstyle></math>, quia uterque <lb/>
est complementum anguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>g</mi></mstyle></math>. et anguli ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> sunt æquales propter similia <lb/>
trianguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi><mi>i</mi></mstyle></math>, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi><mi>i</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Triangles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>d</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi><mi>g</mi></mstyle></math> are equiangular.
For <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>e</mi><mi>g</mi></mstyle></math> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi><mi>d</mi></mstyle></math>, because either is the complement of angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>g</mi></mstyle></math>;
and the angles at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> and 4 d <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>r</mi><mi>e</mi><mi>e</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>l</mi><mi>b</mi><mi>e</mi><mi>c</mi><mi>a</mi><mi>u</mi><mi>s</mi><mi>e</mi><mi>o</mi><mi>f</mi><mi>s</mi><mi>i</mi><mi>m</mi><mi>i</mi><mi>l</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>a</mi><mi>n</mi><mi>g</mi><mi>l</mi><mi>e</mi><mi>s</mi></mstyle></math> adi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>n</mi><mi>d</mi></mstyle></math> bgi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>.</mo></mstyle></math></emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1533" xml:space="preserve">
Non refert an <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> sit <lb/>
minor vel maior quam <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
It does not matter whether <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> is less than or greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f445v" o="445v" n="891"/>
<pb file="add_6782_f446" o="446" n="892"/>
<div xml:id="echoid-div287" type="page_commentary" level="2" n="287">
<p>
<s xml:id="echoid-s1534" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1534" xml:space="preserve">
A continuation from Add MS 6782, f. 445, of Harriot's work on the 'Dati sexti' from Chapter XIX of Viète's
<emph style="it">Variorum responsorum liber VIII</emph> (1593).
Here he examines Proposition I, part 3, which he has divided as 3 and 4,
according to whether one takes the minus or plus sign in the final sentence of the statement.
</s>
<lb/>
<quote xml:lang="lat">
I. <lb/>
Data peripheria composita e duabus peripheriis, quarum transsinuosae datam habeant rationem, dantur singulæ. <lb/>
<lb/>[...]<lb/> <lb/>
3 Et si denique componentium peripheriarum prima fit minor quadrante, secunda major, <lb/>
Erit, <lb/>
Vt transsinuosa primæ ad transsinuosam secundæ, ita transsinuosa complementi compositæ
ad prosinum complementi compositæ minus prosinu secundæ.
Et, <lb/>
Vt transsinuosa secunda ad transsinuousam primæ, ita transsinuosa complementi compositæ
ad prosinum complementi compositæ plus prosinu secundæ.
</quote>
<lb/>
<quote>
I.Given an arc composed of two others, whose secants are in a given ratio, each is known individually. <lb/>
<lb/>[...]<lb/> <lb/>
3. And if further the first arc is less than a quadrant, the second greater, then
as the secant of the first is to the secant of the second, so is the secant of the complement of the sum
to the sine of the complement of the sum minus the sine of the second, <lb/>
and <lb/>
[4.] as the secant of the second to the secant of the first, so is the secant of the complement of the sum
to the sine of the complement of the sum plus the sine of the second.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head199" xml:space="preserve" xml:lang="lat">
Vieta. resp. lib. 8. <lb/>
pag. 38. <lb/>
3. et 4. <lb/>
Data peripheria composita e duabus peripheriis, quarum transsinuosæ <emph style="super"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Î¨</mo></mstyle></math></emph>
datam habeant rationem, dantur singulæ.
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 38, Propositions 3 and 4. <lb/>
Given an arc composed of two arcs, whose secants are in a given ratio, each is given separately.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1536" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math> peripheria composita <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> prima minor quadrante <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> secunda maior quadrante <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi></mstyle></math> Transsinuosa <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, primæ <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>c</mi></mstyle></math> Transsinuosa <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, secundæ
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math> is the sum of the arcs <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> is the first, less than a quadrant <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> is the second, greater than a quadrant <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi></mstyle></math> is the secant of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, the first <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>c</mi></mstyle></math> is the secant of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, the second
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1537" xml:space="preserve">
1<emph style="super">o</emph>. Triangula <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi><mi>g</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi><mi>o</mi></mstyle></math> sunt æquiangula;
propter parallelas <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>o</mi></mstyle></math>. <lb/>
2<emph style="super">o</emph>. Triangula <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>d</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi><mi>m</mi></mstyle></math> sunt æquiangula.
nam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi><mi>d</mi></mstyle></math> est complementum anguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>h</mi></mstyle></math> inde æqualis <lb/>
angulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>k</mi></mstyle></math>, cui æqualis fit per constructionem <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>e</mi><mi>m</mi></mstyle></math>. anguli igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>e</mi><mi>m</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi><mi>d</mi></mstyle></math> sunt æquales <lb/>
et anguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>d</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi><mi>m</mi></mstyle></math> sunt æquales propter parallelas <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>f</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi></mstyle></math>. Tertius ergo angulus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi><mi>g</mi></mstyle></math> <lb/>
est æqualis tertio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>m</mi><mi>e</mi></mstyle></math>. <lb/>
3<emph style="super">o</emph>. Arcus <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>n</mi><mi>k</mi></mstyle></math> sunt æquales. nam arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> est arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi><mo>-</mo><mi>b</mi><mi>k</mi></mstyle></math>,
hoc est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi><mo>-</mo><mi>k</mi><mi>n</mi></mstyle></math>. <lb/>
4<emph style="super">o</emph>. Triangula <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>h</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>e</mi><mi>o</mi></mstyle></math> sunt æqualia et æquiangula.
<lb/>[<emph style="it">tr:
1. Triangles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi><mi>g</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi><mi>o</mi></mstyle></math> are equiangular, because <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>o</mi></mstyle></math> are parallel. <lb/>
2. Triangles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>d</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi><mi>m</mi></mstyle></math> are equiangular, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi><mi>d</mi></mstyle></math> is the complement of angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>h</mi></mstyle></math>,
thence equal to angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>k</mi></mstyle></math>, to which <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>e</mi><mi>m</mi></mstyle></math> is equal by construction;
therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>e</mi><mi>m</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi><mi>d</mi></mstyle></math> are equal,
and angles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>d</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi><mi>m</mi></mstyle></math> are equal because <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>f</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi></mstyle></math> are parallel.
Therefore a third of angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi><mi>g</mi></mstyle></math> is equal to a third of angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>m</mi><mi>e</mi></mstyle></math>. <lb/>
3. Arcs <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>n</mi><mi>k</mi></mstyle></math> are equal, for arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> is arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi><mo>-</mo><mi>b</mi><mi>k</mi></mstyle></math>,
that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi><mo>-</mo><mi>k</mi><mi>n</mi></mstyle></math>. <lb/>
4. Triangles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>h</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>e</mi><mi>o</mi></mstyle></math> are equal and equiangular.<lb/>
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1538" xml:space="preserve">
peripheria <lb/>
una minor <lb/>
altera maior <lb/>
quadrante
<lb/>[<emph style="it">tr:
one of the arcs is less than, the other greater than a quadrant
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f446v" o="446v" n="893"/>
<pb file="add_6782_f447" o="447" n="894"/>
<div xml:id="echoid-div288" type="page_commentary" level="2" n="288">
<p>
<s xml:id="echoid-s1539" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1539" xml:space="preserve">
A continuation from Add MS 6782, f. 445 and f. 446, of Harriot's work on the 'Dati sexti'
from Chapter XIX of Viète's
<emph style="it">Variorum responsorum liber VIII</emph> (1593).
Here he examines Proposition II, part 1.
</s>
<lb/>
<quote xml:lang="lat">
II. <lb/>
Data differentia duarum perpheriarum, quarum transsinuosæ datam habeant rationem, dantur singulæ. <lb/>
1 Enimvero si differentia fit major circuli quadrante, <lb/>
Erit, <lb/>
Vt transsinuosa differentium primæ ad transsinuosum secundæ, ita transsinuosa complementi differentiæ
ad prosinum complementi differentiaæ plus prosinu secundæ.
</quote>
<lb/>
<quote>
II.Given the difference of two arcs, whose secants are in a given ratio, each is given individually. <lb/>
<lb/>[...]<lb/> <lb/>
1. If the difference is greater than a quadrant, then
as the secant of the first is to the secant of the second, so is the secant of the complement of the difference
to the sine of the complement of the difference plus the sine of the second.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head200" xml:space="preserve" xml:lang="lat">
Vieta. resp. lib. 8. <lb/>
pag. 38. <lb/>
Data differentia duarum perpheriarum
et ratione <emph style="super"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Î¨</mo></mstyle></math></emph>
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 38. <lb/>
Given the difference of two arcs, and the ratio of their secants.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1541" xml:space="preserve">
1. <lb/>
differentia <lb/>
maior quadrante
<lb/>[<emph style="it">tr:
1. difference greater than a quadrant.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1542" 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 <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>i</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>i</mi></mstyle></math> differentia <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>k</mi></mstyle></math> compl. differentiæ <lb/>
<lb/>[...]<lb/> <lb/>
Triangula similia
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> one arc <lb/>
<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>b</mi><mi>i</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>i</mi></mstyle></math> the difference <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>k</mi></mstyle></math> complement of the difference <lb/>
<lb/>[...]<lb/> <lb/>
Similar triangles
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1543" xml:space="preserve">
primus terminus minor <lb/>
primus term maior
<lb/>[<emph style="it">tr:
the first term less <lb/>
the first term greater
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f447v" o="447v" n="895"/>
<pb file="add_6782_f448" o="448" n="896"/>
<div xml:id="echoid-div289" type="page_commentary" level="2" n="289">
<p>
<s xml:id="echoid-s1544" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1544" xml:space="preserve">
A continuation of Harriot's work on the 'Dati sexti' from Chapter XIX of Viète's
<emph style="it">Variorum responsorum liber VIII</emph> (1593).
Here he examines Proposition II, part 2.
</s>
<lb/>
<quote xml:lang="lat">
II. <lb/>
Data differentia duarum perpheriarum, quarum transsinuosæ datam habeant rationem, dantur singulæ. <lb/>
<lb/>[...]<lb/> <lb/>
2 Et si differentia fit minor quadrante, differentes autem peripheriæ diversæ sint speciei. <lb/>
Erit, <lb/>
Vt transsinuosa primæ ad transsinuosum secundæ, ita transsinuosa complementi differentiæ
ad prosinum secundæ minus prosinu complementi differentiæ.
</quote>
<lb/>
<quote>
II.Given the difference of two arcs, whose secants are in a given ratio, each is given individually. <lb/>
<lb/>[...]<lb/> <lb/>
2. And if the difference is less than a quadrant, and also the signs of each arc are different, then
as the secant of the first is to the secant of the second, so is the secant of the complement of the difference
to the sine of the second minus the sine of the complement of the difference.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head201" xml:space="preserve" xml:lang="lat">
Vieta. resp. lib. 8. <lb/>
pag. 38. <lb/>
Data differentia duarum perpheriarum
et ratione <emph style="super"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Î¨</mo></mstyle></math></emph>
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VIII, page 38. <lb/>
Given the difference of two arcs, and the ratio of their secants.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1546" xml:space="preserve">
2. <lb/>
Differentia minor <lb/>
quadrante. <lb/>
peripheria <lb/>
una minor, <lb/>
altera maior quadrante
<lb/>[<emph style="it">tr:
2. Difference less than a quadrant; one arc greater than a quadrant, the other less.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1547" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> una peripheria minor quadrante <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> altera peripheria maior quadrante <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>i</mi><mo>=</mo><mi>a</mi><mi>b</mi><mo>=</mo><mi>k</mi><mi>m</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>i</mi></mstyle></math> differentia, minor quadrante <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>n</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>f</mi></mstyle></math> parallelæ <lb/>
Triangula similia
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> one arc, less than a quadrant <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> the other arc, greater than a quadrant <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>i</mi><mo>=</mo><mi>a</mi><mi>b</mi><mo>=</mo><mi>k</mi><mi>m</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>i</mi></mstyle></math> the difference, less than a quadrant <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>n</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>f</mi></mstyle></math> parallels <lb/>
Similar triangles
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f448v" o="448v" n="897"/>
<pb file="add_6782_f449" o="449" n="898"/>
<div xml:id="echoid-div290" type="page_commentary" level="2" n="290">
<p>
<s xml:id="echoid-s1548" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1548" xml:space="preserve">
A continuation of Harriot's work on the 'Dati sexti' from Chapter XIX of Viète's
<emph style="it">Variorum responsorum liber VIII</emph> (1593).
Here he examines Proposition II, part 3.
</s>
<lb/>
<quote xml:lang="lat">
II. <lb/>
Data differentia duarum perpheriarum, quarum transsinuosæ datam habeant rationem, dantur singulæ. <lb/>
<lb/>[...]<lb/> <lb/>
3 Et si denique differentia fit minor quadrante, utraque vero differentium vel minor quadrante,
vel utraque major. Prima autem intelligatur ea ad quam pertinet transsiuosa major. <lb/>
Erit, <lb/>
Vt transsinuosa primæ ad transsinuosum secundæ, ita transsinuosa complementi differentiæ
ad prosinum secundæ minus prosinu complementi differentiæ. <lb/>
Et, <lb/>
Vt transsinuosa primæ ad transsinuosum secundæ, ita transsinuosa complementi differentiæ
ad prosinum primæ minus prosinu complementi differentiæ.
</quote>
<lb/>
<quote>
II.Given the difference of two arcs, whose secants are in a given ratio, each is given individually. <lb/>
<lb/>[...]<lb/> <lb/>
3. If the difference is less than a quadrant, and both arcs are less than a quadrant, or both greater, then
as the secant of the first is to the secant of the second, so is the secant of the complement of the difference
to the sine of the second minus the sine of the complement of the difference; and
as the secant of the first is to the secant of the second, so is the secant of the complement of the difference
to the sine of the first minus the sine of the complement of the difference.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head202" xml:space="preserve" xml:lang="lat">
Vieta. pag. 38.b. resp. lib. 8. <lb/>
Data differentia duarum perpheriarum
et ratione <emph style="super"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Î¨</mo></mstyle></math></emph>
<lb/>[<emph style="it">tr:
Viète, page 38v, Responsorum liber VIII. <lb/>
Given the difference of two arcs, and the ratio of their secants.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1550" xml:space="preserve">
3. bis
<lb/>[<emph style="it">tr:
3. twofold.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1551" 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 <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>i</mi><mo>=</mo><mi>a</mi><mi>b</mi><mo>=</mo><mi>k</mi><mi>m</mi></mstyle></math> <lb/>
<lb/>[...]<lb/> <lb/>
Ergo. anguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>o</mi><mi>g</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>d</mi></mstyle></math> æquales <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>n</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math> parallelæ <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi><mo>=</mo><mi>k</mi><mi>n</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>c</mi></mstyle></math>, differentia inter <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>b</mi><mi>c</mi></mstyle></math> <lb/>
Triangula similia
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> one arc <lb/>
<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>b</mi><mi>i</mi><mo>=</mo><mi>a</mi><mi>b</mi><mo>=</mo><mi>k</mi><mi>m</mi></mstyle></math> <lb/>
<lb/>[...]<lb/> <lb/>
Therefore angles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>o</mi><mi>g</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>d</mi></mstyle></math> are equal <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>n</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math> are parallels <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi><mo>=</mo><mi>k</mi><mi>n</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>c</mi></mstyle></math>, 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>b</mi><mi>c</mi></mstyle></math> <lb/>
Similar triangles
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1552" xml:space="preserve">
Differentia minor <lb/>
quadrante. <lb/>
Utraque peripheria <lb/>
etiam minor,
<lb/>[<emph style="it">tr:
Difference less than a quadrant; each arc also less.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f449v" o="449v" n="899"/>
<pb file="add_6782_f450" o="450" n="900"/>
<div xml:id="echoid-div291" type="page_commentary" level="2" n="291">
<p>
<s xml:id="echoid-s1553" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1553" xml:space="preserve">
The reference on this page is to Viète's
<emph style="it">Variorum responsorum liber VIII</emph>,
Chapter 14, Proposition 1.
</s>
<lb/>
<quote xml:lang="lat">
Propositio I. <lb/>
Datis duabus inæqualibus lineis, una recta, altera circulari, invenire lineam rectam
minorem majore datarum, &amp; majorem minore.
</quote>
<lb/>
<quote>
Given two unequal lines, one straight, the other circular, to find a straight line
less than the greater of those given, and greater than the lesser.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head203" xml:space="preserve" xml:lang="lat">
Resp. lib. 8. cap. 14. prop. 1. pag. 23
<lb/>[<emph style="it">tr:
Responsorum liber VIII, Chapter 14, Proposition 1, page 23.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1555" xml:space="preserve">
Datis duabus inæqualibus lineis: una recta, altera circulari: <lb/>
invenire lineam rectam, minorem majore datarum et majorem minore.
<lb/>[<emph style="it">tr:
Given two unequal lines, one straight, the other circular, to find a straight line
less than the greater of those given, and greater than the lesser.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1556" xml:space="preserve">
Casus primus.
<lb/>[<emph style="it">tr:
First case.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1557" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> recta. maior. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi></mstyle></math> circularis <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math> Excessus <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math> Circulari
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> be the straight line, greater. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi></mstyle></math> the circular line <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math> the excess <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math> the circular line.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1558" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> maior circulari <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> quæcunque maior <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> is greater than the circular line. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> is any quantity greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1559" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>.</mo><mi>G</mi><mo>.</mo><mi>H</mi><mo>-</mo><mi>Z</mi><mo>.</mo><mi>A</mi></mstyle></math> quæsita linea <lb/>
Nam subducendo ab <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>-</mo><mi>Z</mi></mstyle></math>. remanabit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. <lb/>
et a <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>. remanabit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>A</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> igitur est minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>:</mo><mi>G</mi><mo>=</mo><mi>H</mi><mo>-</mo><mi>Z</mi><mo>:</mo><mi>A</mi></mstyle></math>, the line sought. <lb/>
For subtracting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>-</mo><mi>Z</mi></mstyle></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math>, there remains <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>A</mi></mstyle></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>, there remains <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>A</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> is therefore less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1560" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>.</mo><mi>G</mi><mo>.</mo><mi>Z</mi><mo>.</mo><mi>F</mi></mstyle></math> minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. <lb/>
et subducendo: <lb/>
Ergo collata ista proportione cum prima: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>F</mi><mo>=</mo><mi>A</mi></mstyle></math>. <lb/>
Cum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> sit minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>, ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>F</mi></mstyle></math> est Maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math>. <lb/>
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> est Maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math>. et minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>:</mo><mi>G</mi><mo>=</mo><mi>Z</mi><mo>:</mo><mi>F</mi></mstyle></math> less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. <lb/>
and subtracting: <lb/>
Therefore, combining this proportion with the first, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>F</mi><mo>=</mo><mi>A</mi></mstyle></math>. <lb/>
Since <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>, therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>F</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math>. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math> and less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f450v" o="450v" n="901"/>
<pb file="add_6782_f451" o="451" n="902"/>
<div xml:id="echoid-div292" type="page_commentary" level="2" n="292">
<p>
<s xml:id="echoid-s1561" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1561" xml:space="preserve">
This page is a continuation of Add MS 6782, f. 450, on Viète's
<emph style="it">Variorum responsorum liber VIII</emph>,
Chapter 14, Proposition 1.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head204" xml:space="preserve" xml:lang="lat">
Casus primus.
<lb/>[<emph style="it">tr:
First case.
</emph>]<lb/>
</head>
<pb file="add_6782_f451v" o="451v" n="903"/>
<pb file="add_6782_f452" o="452" n="904"/>
<div xml:id="echoid-div293" type="page_commentary" level="2" n="293">
<p>
<s xml:id="echoid-s1563" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1563" xml:space="preserve">
This page is a continuation of Add MS 6782, f. 450 and f. 451, on Viète's
<emph style="it">Variorum responsorum liber VIII</emph>,
Chapter 14, Proposition 1.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head205" xml:space="preserve" xml:lang="lat">
Resp. lib. 8. cap. 14. prop. 1. pag. 23
<lb/>[<emph style="it">tr:
Responsorum liber VIII, Chapter 14, Proposition 1, page 23.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1565" xml:space="preserve">
Casus secundus.
<lb/>[<emph style="it">tr:
Second case.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1566" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> recta. minor. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi></mstyle></math> circularis <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math> Excessus
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> be the straight line, lesser. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi></mstyle></math> the circular line <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math> the excess
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1567" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> minor circulari
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> is less than the circular line.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1568" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>.</mo><mi>G</mi><mo>.</mo><mi>H</mi><mo>+</mo><mi>Z</mi><mo>.</mo><mi>E</mi></mstyle></math> quæsita linea <lb/>
Et subducendo ab <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>+</mo><mi>Z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math>: remanabit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. <lb/>
et ab <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>G</mi></mstyle></math>. remanabit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mo>-</mo><mi>G</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> igitur maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>:</mo><mi>G</mi><mo>=</mo><mi>H</mi><mo>+</mo><mi>Z</mi><mo>:</mo><mi>E</mi></mstyle></math>, the line sought. <lb/>
For subtracting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>+</mo><mi>Z</mi></mstyle></math>, there remains <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>G</mi></mstyle></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math>, there remains <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mo>-</mo><mi>G</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> is therefore greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1569" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>.</mo><mi>G</mi><mo>.</mo><mi>Z</mi><mo>.</mo><mi>F</mi></mstyle></math> minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. <lb/>
Et per additione: <lb/>
Ergo, collata ista proportione cum prima: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>+</mo><mi>F</mi><mo>=</mo><mi>E</mi></mstyle></math>. <lb/>
Cum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> sit minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>, ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>+</mo><mi>F</mi></mstyle></math> est minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>+</mo><mi>Z</mi></mstyle></math>. <lb/>
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> est minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>+</mo><mi>Z</mi></mstyle></math>. et Maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>:</mo><mi>G</mi><mo>=</mo><mi>Z</mi><mo>:</mo><mi>F</mi></mstyle></math> less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. <lb/>
And by addition: <lb/>
Therefore, combining this proportion with the first, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>+</mo><mi>F</mi><mo>=</mo><mi>E</mi></mstyle></math>. <lb/>
Since <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>, therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>+</mo><mi>F</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>+</mo><mi>Z</mi></mstyle></math>. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math> and greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f452v" o="452v" n="905"/>
<pb file="add_6782_f453" o="453" n="906"/>
<div xml:id="echoid-div294" type="page_commentary" level="2" n="294">
<p>
<s xml:id="echoid-s1570" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1570" xml:space="preserve">
This page appears to be a continuation of Harriot's work on Add MS 6782, f. 450 and f. 452,
on Proposition 1 from Viète's <emph style="it">Variorum responsorum liber VIII</emph>, Chapter 14.
Here he also refers to Euclid's <emph style="it">Elements</emph>,
Book III, Proposition 16.
</s>
<lb/>
<quote>
III.16 The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle,
and into the space between the straight line and the circumference another straight line cannot be interposed;
further the angle of the semicircle is greater, and the remaining angle less, than any acute rectilineal angle.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s1572" xml:space="preserve">
Si angulus semicirculi sit minor recto rectilineo: dabitur <lb/>
angulus rectilineus maior angulo semicirculi et minor recto <lb/>
rectilineo. Contra Eculidem lib. 3. prop. 16.
<lb/>[<emph style="it">tr:
If the angle in a semicircle is less than a right angle,
there may be found an angle greater than the angle in the semicircle and less than a right angle.
Against Euclid III.16.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1573" xml:space="preserve">
Si angulus rectus et maior angulo semicirculi. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. <lb/>
Differentia inter angulum rectum et angulum semicurculi. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. <lb/>
Ergo angulus semicirculi. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math>. <lb/>
Et sit aliquis angulus rectilineus maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>H</mi></mstyle></math>
<lb/>[<emph style="it">tr:
Suppose a right angle is greater than the angle in a semicircle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. <lb/>
The difference between the right angle and the angle in the semicircle is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. <lb/>
Therefore the angle in the semicircle is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math>. <lb/>
And let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> be any angle greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1574" xml:space="preserve">
Tum ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> ita <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>-</mo><mi>Z</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>.
aliquem <emph style="super">rectilineum</emph> angulum ... qui <lb/>
minor est quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> et maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Then as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> is to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>-</mo><mi>Z</mi></mstyle></math> is to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>, another angle which is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> and greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1575" xml:space="preserve">
Nam subducendo ab <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>-</mo><mi>Z</mi></mstyle></math>: remanebit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math> <lb/>
et ab <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>: remanebit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>A</mi></mstyle></math>. <lb/>
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>.</mo><mi>G</mi><mo>.</mo><mi>Z</mi><mo>.</mo><mi>G</mi><mo>-</mo><mi>A</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> igitur minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. <lb/>
Et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>.</mo><mi>G</mi><mo>.</mo><mi>Z</mi><mo>.</mo><mi>F</mi></mstyle></math> minorem quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. <lb/>
Ergo subducendo erit <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>.</mo><mi>G</mi><mo>.</mo><mi>H</mi><mo>-</mo><mi>Z</mi><mo>.</mo><mi>G</mi><mo>-</mo><mi>F</mi></mstyle></math> <lb/>
Ergo collata ista proportione cum prima: erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>F</mi><mo>=</mo><mi>A</mi></mstyle></math>. <lb/>
Et: cum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> sit minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>, Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>F</mi></mstyle></math> est Maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math>. <lb/>
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> est Maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math> et minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Now subtracting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>-</mo><mi>Z</mi></mstyle></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> there remains <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>A</mi></mstyle></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>, there remains <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>A</mi></mstyle></math>. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>:</mo><mi>G</mi><mo>=</mo><mi>Z</mi><mo>:</mo><mi>G</mi><mo>-</mo><mi>A</mi></mstyle></math>, therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. <lb/>
And <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>:</mo><mi>G</mi><mo>=</mo><mi>Z</mi><mo>:</mo><mi>F</mi></mstyle></math> less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. <lb/>
Therefore, subtracting, <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>:</mo><mi>G</mi><mo>=</mo><mi>H</mi><mo>-</mo><mi>Z</mi><mo>:</mo><mi>G</mi><mo>-</mo><mi>F</mi></mstyle></math> <lb/>
Therefore, combining this proportion with the first, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>F</mi><mo>=</mo><mi>A</mi></mstyle></math>. <lb/>
And since <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>, therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>F</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math>. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math> and less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f453v" o="453v" n="907"/>
<pb file="add_6782_f454" o="454" n="908"/>
<div xml:id="echoid-div295" type="page_commentary" level="2" n="295">
<p>
<s xml:id="echoid-s1576" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1576" xml:space="preserve">
The reference on this page is to Viète's
<emph style="it">Variorum responsorum liber VIII</emph>,
Chapter 14, Proposition 3.
</s>
<lb/>
<quote xml:lang="lat">
Propositio III. <lb/>
Circulo dato, &amp; linea recta in eo inscripta, quæ diametro minor existat,
&amp; alia insuper quæ circulum tangat in inscriptæ termino, educere lineam e centro
ita secantem circulum &amp; ipsam tangentem, ut pars educate e centro interjacens
inter cirumferentiam &amp; inscriptam se habeat ad partem tangentis quæ est inter contactum &amp; ipsam eductam,
sicut dimidia inscripta ad majorem ea quæ ex centro inscriptam illam bisariam secat.
</quote>
<lb/>
<quote>
Given a circle and a straight line inscribed in it, less than the diameter,
and also a line that touches the circle at the end of the inscribed line,
to draw a line from the centre cutting the circle and the tangent, so that the part drawn from the centre
lying between the circumference and the inscribed line is to the part of the tangent
between the contact and the drawn line as half the inscribed line to a line longer than that
drawn from the centre bisecting the inscribed line.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head206" xml:space="preserve" xml:lang="lat">
In Caput 14. Responsorum vieta prop. 3. pa. 24
<lb/>[<emph style="it">tr:
From Chapter 14 of Viète's Responsorum, Proposition 3, page 24.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1578" xml:space="preserve">
et subducendo:
<lb/>[<emph style="it">tr:
and subtracting:
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f454v" o="454v" n="909"/>
<pb file="add_6782_f455" o="455" n="910"/>
<div xml:id="echoid-div296" type="page_commentary" level="2" n="296">
<p>
<s xml:id="echoid-s1579" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1579" xml:space="preserve">
The reference on this page is to Viète's
<emph style="it">Variorum responsorum liber VIII</emph>, Chapter 9, Proposition 13.
</s>
<lb/>
<quote xml:lang="lat">
Et si fuerint lineæ quotcunque sese excedentes, fit autem prima excessui aequalis,
fiunt ab iis quatuor solida coninue proportionalia, qualia sequntur. <lb/>
Primum, Cubus minimæ. <lb/>
Secundum, Cubus compositæ ex maxima &amp; minima, multaus adgregato cuborum minimæ &amp; maximæ. <lb/>
Tertium, Adgregatum cuborume singulis ter duodecuplum. <lb/>
Quartum, Cubus compositæ ex omnibus sextuplae.
</quote>
<lb/>
<quote>
And if there are any number of lines exceeding each other, and moreover the first differences are equal,
there arise from them four solids in continued proportion, which are as follows. <lb/>
First, the cube of the least. <lb/>
Second, the cube of the sum of the greatest and least, reduced by the sum of the cubes of the least and the greatest. <lb/>
Third, the sum of the cubes of each taken 36 times. <lb/>
Fourth, the cube of the sum of all, taken six times.
</quote>
<lb/>
<s xml:id="echoid-s1580" xml:space="preserve">
Note that in his initial calculations, expressed in terms of geometric solids,
Harriot retains homogeneity by writing, for example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>1</mn></mstyle></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>×</mo><mn>1</mn></mstyle></math>.
In the second column, where he moves from a geometric to arithmetic interpretation,
he simplifies the notation by dropping the 1s.
</s>
<lb/>
<s xml:id="echoid-s1581" xml:space="preserve">
For Harriot's teaching on progressions, see in particular Add MS 6782, f. 107 to f. 146v.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head207" xml:space="preserve" xml:lang="lat">
In propositione 13, cap.9. lib. 8. respons. pag. 14. b. Vietæ
<lb/>[<emph style="it">tr:
From Proposition 13, Chapter 9, Liber VIII responsorum, page 14v, of Viète
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1583" xml:space="preserve">
propositio <lb/>
Si fuerint lineæ quotcunque sese excedentes, fit autem prima excessui aequalis, <lb/>
fiunt ab iis quatuor solida coninue proportionalia, qualia sequntur. <lb/>
Primum, Cubus minimæ. <lb/>
Secundum, Cubus compositæ ex maxima et minima, multaus aggregato <lb/>
cuborum minimæ et maximæ. <lb/>
Tertium, Adgregatum cuborume singulis ter duodecuplum. <lb/>
Quartum, Cubus compositæ ex omnibus sextuplae.
<lb/>[<emph style="it">tr:
If there are any number of lines exceeding each other, and moreover the first differences are equal,
there arise from them four solids in continued proportion, which are as follows. <lb/>
First, the cube of the least. <lb/>
Second, the cube of the sum of the greatest and least, reduced by the sum of the cubes of the least and the greatest. <lb/>
Third, the sum of the cubes of each taken 36 times. <lb/>
Fourth, the cube of the sum of all, taken six times.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1584" xml:space="preserve">
Sit minima 1. maxima 4. Summa omnia sextupla sit 60. et sunt continuo <lb/>
proportionalia solida. 1. 60. 360. 21600.
<lb/>[<emph style="it">tr:
Let the least quantity be 1, the greatest 4. Six times the sum of all of them is 60,
and the proportional solids are 1, 60, 360, 21600.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1585" xml:space="preserve">
Sit prima linea et excessus 1. <lb/>
et numerus linearum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> <lb/>
Summa <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>, sive triangulus numerus <lb/>
ipsius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi><mi>n</mi><mo>+</mo><mn>1</mn><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>. <lb/>
cuius quadratum erit: <lb/>
atque hoc est per doctrina progressionum <lb/>
summa cuborum singulis <lb/>
Eis ter duodecuplum erit: <lb/>
Hoc est: <lb/>
pro 3<emph style="super">o</emph> solido
<lb/>[<emph style="it">tr:
Let the first line and the excess be 1, and <lb/> the number of lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>. <lb/>
Their sum, or triangular number, will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi><mi>n</mi><mo>+</mo><mn>1</mn><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>, whose square is: <lb/>
and by the doctrine of progressions, this is the sum of individual cubes. <lb/>
Three times their 12-tuple will be: <lb/>
That is: <lb/>
for the 3rd solid.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1586" xml:space="preserve">
Composita ex maxima et minima <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>+</mo><mn>1</mn></mstyle></math> <lb/>
Eius cubus erit <lb/>
Cubis multatis; erit <lb/>
pro 2<emph style="super">o</emph> solido
<lb/>[<emph style="it">tr:
The sum of the greatest and least is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>+</mo><mn>1</mn></mstyle></math>. <lb/>
Its cube will be: </emph>]<lb/>
The cubes having been subtracted this will be: <lb/>
for the 2nd solid.
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1587" xml:space="preserve">
Composita ex omnibus ut supra <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi><mi>n</mi><mo>+</mo><mn>1</mn><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> <lb/>
Eius sextupla: <lb/>
Hoc est: <lb/>
Eius quadratum: <lb/>
Eius cubus: <lb/>
pro 4<emph style="super">o</emph> <lb/>
solido <lb/>
Quadratum si <emph style="st">dividatur</emph> <emph style="super">multiplicatur</emph> per 1.
<emph style="st">[???]</emph> <emph style="super">primam lineam</emph> <lb/>
faciet <emph style="st">secundum</emph> <emph style="super">tertium</emph> solidum ut supra.
<lb/>[<emph style="it">tr:
The sum of all, as above, is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi><mi>n</mi><mo>+</mo><mn>1</mn><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>. <lb/>
Six time this: <lb/>
That is: <lb/>
Its square: <lb/>
Its cube: <lb/>
for the 4th solid. <lb/>
The square, if the first line is multiplied by 1, makes the third solid, as above.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1588" xml:space="preserve">
Solida continue proportionales
<lb/>[<emph style="it">tr:
The continually proportional solids
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1589" xml:space="preserve">
proportionalis operatio, dato primo <lb/>
et secundo, inveniet 3<emph style="super">um</emph> et 4<emph style="super">tum</emph> <lb/>
sub maiori formam notari sed <lb/>
per reductionem; æqualia sunt illis.
<lb/>[<emph style="it">tr:
The operation of proportion, given the first and second, will find the 3rd and 4th, under a greater form of notation,
but by reduction, they are equal.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1590" xml:space="preserve">
pro numeris vel numeris <lb/>
solidarum talis potest <lb/>
esse notatio: et si fit <lb/>
in forma heterogenea. Vel: <lb/>
ennuntiari potest per lineas.
<lb/>[<emph style="it">tr:
for numbers, or solid numbers, the notation my be thus; and if done in heteregenous form.
Or it may be expressed in lines.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1591" xml:space="preserve">
datis primo et secundo
proportionalis operatio 3<emph style="super">um</emph> &amp; 4<emph style="super">um</emph> <lb/>
invenient ut sunt.
<lb/>[<emph style="it">tr:
given the first and second, by the operation of proportions, the 3rd and 4th may be found and are:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1592" xml:space="preserve">
Nota, quod <lb/>
etiam proportio potest <lb/>
inveniri per plana <lb/>
per lineas <lb/>
per plano-plana &amp;c. <lb/>
et eadem modi demonstrari. Etiam: <lb/>
proportionalia possunt fieri plano numero <lb/>
ad libitum secundum nostram doctrinam de <lb/>
progressionibus, quæ per traditiones veterum <lb/>
fieri non potuit.
<lb/>[<emph style="it">tr:
Note, that the proportion may also be found by planes, lines, plano-planes, etc.
and demosntrated by the same method. Also, the proportionals may arise from plane numbers at will
according to my doctrine of progressions, which by old teachings could not be done.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f455v" o="455v" n="911"/>
<pb file="add_6782_f456" o="456" n="912"/>
<div xml:id="echoid-div297" type="page_commentary" level="2" n="297">
<p>
<s xml:id="echoid-s1593" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1593" xml:space="preserve">
This page contains Harriot's working of Zetetic 10, the last from Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book I.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum X <lb/>
Invenire duo latera, quorum differentia fit ea quæ præscribitur,
&amp; præterea præfinitae unciæ primi, multatæ præfinitis unciis secundi,
æquent differentiam quoque inter eas datam.
</quote>
<lb/>
<quote>
To find two lines, whose difference is prescribed,
and also such that a fixed part of the first taken from a fixed part of the second
is likewise equal to a given difference.
</quote>
<lb/>
<s xml:id="echoid-s1594" xml:space="preserve">
Viète used the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the difference between the two lines,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the first part to the first line,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the second part to the second line,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> for the gvien difference,
and <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> for the parts of the first and second lines.
Harriot worked through the three cases given by Viète, using his own notation. <lb/>
The work is continued using numbers on Add MS 6782, f. 459.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head208" xml:space="preserve" xml:lang="lat">
Zetet. lib. 1. Zet. 10. et primi lib. ultimum	Sept. 6.
<lb/>[<emph style="it">tr:
Zetetica, Book I, Zetetic 10, and the last of the first book. September 6
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1596" xml:space="preserve">
Invenire duo latera, quorum differentia sit ea quæ præscribitur, et præterea <lb/>
præfinitæ unciæ lateris primi multatæ præfinitis unciis secundi, æquent <lb/>
differentiam quoque inter eas datam.
<lb/>[<emph style="it">tr:
To find two lines, whose difference is prescribed,
and also such that a fixed part of the first subtracted from a fixed part of the second
is likewise equal to a given difference.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1597" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. data differentia 2<emph style="super">orum</emph> laterum. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. primum latus. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. secundum latus. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. portio primi lateri. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math> <lb/>
hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>a</mi><mo>,</mo><mi>o</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. portio secundi lateris <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>e</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math> <lb/>
hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>e</mi><mo>,</mo><mi>u</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. differentia data <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>a</mi><mo>-</mo><mi>e</mi></mstyle></math> <lb/>
unde: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>+</mo><mi>e</mi><mo>=</mo><mi>a</mi></mstyle></math>. et: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>h</mi><mo>=</mo><mi>e</mi></mstyle></math>. <lb/>
Quæruntur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, <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>.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, the given difference between the two lines. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, the first line <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, the second line. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, the portion of the first line. <lb/>
The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math>, <lb/>
that is: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>a</mi><mo>:</mo><mi>o</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, the portion of the second line <lb/>
The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>e</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math>, <lb/>
that is: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>e</mi><mo>:</mo><mi>u</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>, the given difference, equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>e</mi></mstyle></math>. <lb/>
whence: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>+</mo><mi>e</mi><mo>=</mo><mi>a</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>h</mi><mo>=</mo><mi>e</mi></mstyle></math>. <lb/>
There are sought <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, <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>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1598" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. primum latus intelligitur maius duarum <lb/>
vel minus; sive ab eo exigantur unciæ <lb/>
maioris vel minoris.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, the first line can be understood to be the greater of the two, <lb/>
or the smaller; whether from it are taken greater fractions or smaller.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1599" xml:space="preserve">
1. casus. sit primum latus maius: <lb/>
et exigantur ab eo maioris unciæ. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> portio primi lateris. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math> portio secundi lateris.
<lb/>[<emph style="it">tr:
Case 1. Let the first line be greater, and from it are taken greater fractions. <lb/>
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be the portion of the first line, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math> the portion of the second line.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1600" xml:space="preserve">
PorrÃ², sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, portio <lb/>
secundi lateris.
<lb/>[<emph style="it">tr:
Further, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the portion of the second line.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1601" xml:space="preserve">
2. casus. sit primum latus maius <lb/>
et exigantur ab eo minoris unciæ. <lb/>
hoc est ponatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Case 2. Let the first line be greater, and from it are taken smaller fractions,
that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is supposed less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1602" xml:space="preserve">
PorrÃ², sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, portio <lb/>
secundi lateris.
<lb/>[<emph style="it">tr:
Further, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the portion of the second line.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1603" xml:space="preserve">
Nota. <lb/>
Hinc apparet quod <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> debet esse. <lb/>
In primo casu maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, ac etiam quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> <lb/>
In secundo casu minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. <lb/>
In tertio maior vel minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Note. <lb/>
Here it is clear what <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> must be. <lb/>
In the first case, greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> and also than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. <lb/>
In the second case, less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. <lb/>
In the third, greater or less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1604" xml:space="preserve">
3. casus. sit primum latus minus <lb/>
et exigantur ab eo maioris unciæ. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> portio primi lateris. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math> portio secundi lateris.
<lb/>[<emph style="it">tr:
Case 3. Let the first line be smaller, and from it ar taken greater fractions. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> is the portion of the first line, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math> the portion of the second line.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1605" xml:space="preserve">
PorrÃ², sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, portio secundi lateris.
<lb/>[<emph style="it">tr:
Further, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the portion of the second line.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1606" xml:space="preserve">
* Nota. <lb/>
Hinc facile apparet quod minores <lb/>
unciæ non possunt exigi Ã  primo latere <lb/>
cum sit minus. Atque ideo non <lb/>
datur quartus casus.
<lb/>[<emph style="it">tr:
* Note. <lb/>
Here is is easily seen that smaller fractions cannot be taken from the first line since it is smaller.
And therefore therefore no fourth case is given.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f456v" o="456v" n="913"/>
<pb file="add_6782_f457" o="457" n="914"/>
<div xml:id="echoid-div298" type="page_commentary" level="2" n="298">
<p>
<s xml:id="echoid-s1607" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1607" xml:space="preserve">
This page contains the continuation of Harriot's working of Zetetic 7 from Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book I.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum VII <lb/>
Datum latus ita secare, ut praefinitae unciae unius segmenti, adjunctae praefinitis unciis alterius:
aequent summam praescriptam.
</quote>
<lb/>
<quote>
To cut a given line in such a way that a fixed part of one segment, added to a fixed part of the other,
is equal to a prescribed sum.
</quote>
<lb/>
<s xml:id="echoid-s1608" xml:space="preserve">
Viète used the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the whole line,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the first part to the first segment,
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the second part to the second segment,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> for the prescribed sum,
and <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> for the unknown parts of the first and second segments.
Harriot followed Viète's method on Add MS 6782, f. 458.
Here he works the same problem using numbers instead of lengths.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head209" xml:space="preserve" xml:lang="lat">
Zet. lib. 1. Zet. 7.
<lb/>[<emph style="it">tr:
Zetetica, Book I, Zetetic 7
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1610" xml:space="preserve">
Dividere numerum <emph style="super">datum</emph> in duas partes, ita ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>
<emph style="super">unius</emph> primæ partis additæ, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math>, secundæ: æquet <lb/>
summam præscriptam. oportuit ut summa præscripta sit
minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> numeri dati et maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math>.
<lb/>[<emph style="it">tr:
To divide a given number into two parts, so that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of one, the first part,
added to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math> of the second equals a prescribed sum;
the prescribed sum must be less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of the given number and greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1611" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. sit numerus datus. 60 <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. igitur 15. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> erit 10. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> summa præscribenda debet <lb/>
esse maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, et minor <lb/>
quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. sit ergo 12.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the given number, 60. Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is 15 and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> will be 10. <lb/>
The prescribed sum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> must be grater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> and less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, therefore let it be 12.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1612" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. prima pars. <lb/>
<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><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> primæ partis.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> be the first part, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> a <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of the first part.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1613" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. secunda pars. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>: eius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math>
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math> be the second part, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> a <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math> of it.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1614" xml:space="preserve">
hac analogia ita solvitur problema. <lb/>
<lb/>[...]<lb/>
Ergo. 36. erit secunda pars. <lb/>
6. erit eius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math>. <lb/>
6. est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mi>p</mi><mi>r</mi><mi>i</mi><mi>m</mi><mo>æ</mo></mstyle></math> <lb/>
quæ æquat 12.
<lb/>[<emph style="it">tr:
The problem is thus solved by this ratio. <lb/>
<lb/>[...]<lb/> <lb/>
Therefore 36 will be the second part. <lb/>
6 is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math> of it, <lb/>
6 is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of the first part <lb/>
which makes 12.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1615" xml:space="preserve">
aliter. <lb/>
<lb/>[...]<lb/>
Ergo. 24. erit prima pars. <lb/>
6. eius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>. <lb/>
6. est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac><mi>s</mi><mi>e</mi><mi>c</mi><mi>u</mi><mi>n</mi><mi>d</mi><mo>æ</mo></mstyle></math>
<lb/>[<emph style="it">tr:
Another way <lb/>
<lb/>[...]<lb/> <lb/>
Therefore 24 will be the first part. <lb/>
6 is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of it, <lb/>
6 is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math> of the second part
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1616" xml:space="preserve">
Aliud exemplum
<lb/>[<emph style="it">tr:
Another example
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f457v" o="457v" n="915"/>
<pb file="add_6782_f458" o="458" n="916"/>
<div xml:id="echoid-div299" type="page_commentary" level="2" n="299">
<p>
<s xml:id="echoid-s1617" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1617" xml:space="preserve">
This is the first of several pages in which Harriot worked on Zetetica 7 to 10 from Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book I.
Harriot at first treated the problems exactly as Viète had done, using the geometric language in which the word
<foreign xml:lang="lat">latus</foreign> represents an unknown line, side, or root.
However, in each case Harriot then switched from geometry to arithmetic,
treating the known and unkown quantities as numbers rather than geometrical quantities.
</s>
<lb/>
<s xml:id="echoid-s1618" xml:space="preserve">
This page contains Harriot's working of Zetetic 7.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum VII <lb/>
Datum latus ita secare, ut præfinitæ unciæ unius segmenti, adjunctae præfinitis unciis alterius:
æquent summam præscriptam.
</quote>
<lb/>
<quote>
To cut a given line in such a way that a fixed part of one segment, added to a fixed part of the other,
is equal to a prescribed sum.
</quote>
<lb/>
<s xml:id="echoid-s1619" xml:space="preserve">
Viète used the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the whole line,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the first part to the first segment,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the second part to the second segment,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> for the prescribed sum,
and <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> for the parts of the first and second segments.
Harriot repeated Viète's working in his own notation, and also added some variants of his own.
The numbers at the bottom of the page are taken from Viète.
On Add MS 6782, f. 457, Harriot worked the same problem with further numerical examples.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head210" xml:space="preserve" xml:lang="lat">
Lib. 1. Zetet. <lb/>
Zet. 7.
<lb/>[<emph style="it">tr:
Zetetica, Book I, Zetetic 7
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1621" xml:space="preserve">
Datum latus ita secare, ut præfinitæ unciæ unius segmenti, adjunctae præfinitis unciis alterius:
æquent summam præscriptam.
<lb/>[<emph style="it">tr:
To cut a given line in such a way that a fixed part of one segment, added to a fixed part of the other,
is equal to a prescribed sum.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1622" xml:space="preserve">
sit datum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. latus. <lb/>
et duo segmenta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math> <lb/>
portio primæ segmenti <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> <lb/>
ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>: ita debet esse: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> <lb/>
summa præscripta sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> <lb/>
Ergo portio 2<emph style="super">i</emph> segmenti erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math>. <lb/>
ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>: ita debet esse: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. <lb/>
Quæretur iam: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, et, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, segmenta <lb/>
et segmentorum portiones.
<lb/>[<emph style="it">tr:
Let the given line be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, and the two segments <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>u</mi></mstyle></math>. <lb/>
The portion of the first segment is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>; as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, so must be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. <lb/>
The prescribed sum is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. <lb/>
Therefore the portion of the second segment will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math>. <lb/>
As <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> is to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, so must be % h - a <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi><mi>o</mi></mstyle></math> u <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>.</mo></mstyle></math><lb/>
There are now sought <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>u</mi></mstyle></math>, the segments, and the portions of the segments.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1623" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> minor <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> et maior <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> and greater then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1624" xml:space="preserve">
Aliter. <lb/>
Sit portio secundæ segmenti, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> <lb/>
Ergo portio primæ segmenti <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>e</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Another way. <lb/>
Let the portion of the second segment be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. <lb/>
Therefore the portion of the first segment is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>e</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1625" xml:space="preserve">
Dantur etiam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, per superiores analogias.
<lb/>[<emph style="it">tr:
<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>u</mi></mstyle></math> are also given by the above ratios.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1626" xml:space="preserve">
Additio nostra. Aliter.
<lb/>[<emph style="it">tr:
An addition of my own. Another way.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1627" xml:space="preserve">
Aliter.
<lb/>[<emph style="it">tr:
Another way.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1628" xml:space="preserve">
Nota. <lb/>
Etsi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, sit maior <emph style="super">quam</emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>; est [???] <lb/>
minor <emph style="super">quam</emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> per suppositione: <lb/>
Sed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> est præscribenda ut <lb/>
sit minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, et maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. <lb/>
ut apparet per inventis analogijs.
<lb/>[<emph style="it">tr:
Note. <lb/>
Although <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, it is [???] less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> by supposition. <lb/>
But <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is prescribed so that it is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> and greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>,
as is clear from the ratios found.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f458v" o="458v" n="917"/>
<pb file="add_6782_f459" o="459" n="918"/>
<div xml:id="echoid-div300" type="page_commentary" level="2" n="300">
<p>
<s xml:id="echoid-s1629" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1629" xml:space="preserve">
This page contains Harriot's working of Zetetic 10, the last from Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book I.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum X <lb/>
Invenire duo latera, quorum differentia fit ea quæ præscribitur,
&amp; præterea præfinitae unciæ primi, multatæ præfinitis unciis secundi,
æquent differentiam quoque inter eas datam.
</quote>
<lb/>
<quote>
To find two lines, whose difference is prescribed,
and also such that a fixed part of the first taken from a fixed part of the second
is likewise equal to a given difference.
</quote>
<lb/>
<s xml:id="echoid-s1630" xml:space="preserve">
Viète used the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the difference between the two lines,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the first part to the first line,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the second part to the second line,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> for the gvien difference,
and <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> for the parts of the first and second lines. <lb/>
Harriot followed Viète's method on Add MS 6782, f. 456.
Here he works the same problem using numbers instead of lengths.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head211" xml:space="preserve" xml:lang="lat">
Zetet. lib. 1.	Zet. 10. et ultimum. <lb/>
Exempla in numeris.
<lb/>[<emph style="it">tr:
Zetetica, Book I, Zetetic 10, and the last. <lb/>
Examples in numbers.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1632" xml:space="preserve">
1. Casus.
<lb/>[<emph style="it">tr:
Case 1.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1633" xml:space="preserve">
3. Casus. Ubi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> est maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Case 3. Where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1634" xml:space="preserve">
3. Casus. Ubi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> est minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Case 3. Where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1635" xml:space="preserve">
2. Casus.
<lb/>[<emph style="it">tr:
Case 2.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1636" xml:space="preserve">
Additio. 3. Casus. Ubi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> et minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
An addition. Case 3. Where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> and less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1637" xml:space="preserve">
Si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> sit æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> non variat casum.
<lb/>[<emph style="it">tr:
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> the case does not change.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f459v" o="459v" n="919"/>
<pb file="add_6782_f460" o="460" n="920"/>
<div xml:id="echoid-div301" type="page_commentary" level="2" n="301">
<p>
<s xml:id="echoid-s1638" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1638" xml:space="preserve">
This page contains Harriot's working of Zetetic 9 from Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book I.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum IX <lb/>
Invenire duo latera, quorum differentia sit ea quæ præscribitur,
&amp; præterea præfinitæ unciæ lateris unius, adjectæ præfinitis unciis alterius,
æquabunt summam præscriptam.
</quote>
<lb/>
<quote>
To find two lines, whose difference is prescribed,
and also such that a fixed part of one line added to a fixed part of the other is equal to a prescibed sum.
</quote>
<lb/>
<s xml:id="echoid-s1639" xml:space="preserve">
Viète used the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the difference between the two lines,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the first part to the first line,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the second part to the second line,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> for the prescribed sum,
and <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> for the parts of the first and second lines.
Harriot repeated Viète's working in his own notation, and also added some variants of his own. <lb/>
The work is continued using numerical examples on Add MS 6782, f. 461.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head212" xml:space="preserve" xml:lang="lat">
Zetet. Lib. 1. Zet. 9.
<lb/>[<emph style="it">tr:
Zetetica, Book I, Zetetic 9
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1641" xml:space="preserve">
Invenire duo latera, quorum differentia sit ea quæ præscribitur, et præterea <lb/>
præfinitæ unciæ lateris unius adjectæ præfinitis unciis alterius: æquabunt <lb/>
summam præscriptam.
<lb/>[<emph style="it">tr:
To find two lines, whose difference is prescribed,
and also such that a fixed part of one line added to a fixed part of the other is equal to a prescribed sum.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1642" xml:space="preserve">
Nota. <lb/>
Summa præscripta <lb/>
videlicet <emph style="st">debe</emph><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> debet <lb/>
esse maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, ut <lb/>
per postremis <emph style="st">notat</emph> infra <lb/>
notatis analogijs appa-<lb/>
rebit <emph style="super">scilicet</emph> in primo casu. <lb/>
In secundo casu <lb/>
primum latus ponitur <lb/>
minus, oportet ut <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> fit maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Note. <lb/>
The prescribed sum, namely <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>, must be grater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>,
as is apparent from the ratio written afterwards below in the first case. <lb/>
In the second case, where the first line is supposed smaller, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> must be greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1643" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. data differentia 2<emph style="super">orum</emph> laterum. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. primum latus. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. secundum latus. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. portio primi lateri. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math> <lb/>
Hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>a</mi><mo>,</mo><mi>o</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. portio secundi lateris <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>e</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math> <lb/>
Hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>e</mi><mo>,</mo><mi>u</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. Summa præscripta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>a</mi><mo>+</mo><mi>e</mi></mstyle></math> <lb/>
Quæruntur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>: et <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>.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, the given difference between the two lines. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, the first line <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, the second line. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, the portion of the first line. <lb/>
The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math> <lb/>
That is: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>a</mi><mo>:</mo><mi>o</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, the portion of the second line <lb/>
The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>e</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math> <lb/>
That is: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>e</mi><mo>:</mo><mi>u</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>, the prescribed sum, equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>+</mo><mi>e</mi></mstyle></math>. <lb/>
There are sought <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>u</mi></mstyle></math>, and <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>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1644" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. primum latus intelligitur maius vel minus. <lb/>
primo casu intelligitur maius.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, the first line can be understood to greater or smaller. <lb/>
In the first case it is understood to be greater.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1645" xml:space="preserve">
Tum: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> portio primi lateris. <lb/>
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math> portio secundi lateris.
<lb/>[<emph style="it">tr:
Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> is the portion of the first line. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math> is the portion of the second line.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1646" xml:space="preserve">
PorrÃ², sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, portio <lb/>
secundi lateris.
<lb/>[<emph style="it">tr:
Further, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> is the portion of the second line.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1647" xml:space="preserve">
Secundo casu primum segmentum <lb/>
intelligitur minus. <lb/>
Ergo secundi segmenti erit maius. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> portio secundi lateris. <lb/>
ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>e</mi></mstyle></math> portio primi lateris, et minoris.
<lb/>[<emph style="it">tr:
In the second case the first line is understood to be smaller. <lb/>
Therefore the second line will be greater. <lb/>
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the portion of the second line. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>e</mi></mstyle></math> is the portion of the first line, and smaller.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1648" xml:space="preserve">
PorrÃ², sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, portio <lb/>
primi lateris minoris.
<lb/>[<emph style="it">tr:
Further, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be the portion of the first, smaller line. </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1649" xml:space="preserve">
Additio nostra pro <lb/>
secundus casus aliter. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, portio primi lateris et minoris <lb/>
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math> portio secundi lateris, et maioris. <lb/>
Quoniam: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>a</mi><mo>,</mo><mfrac><mrow><mi>b</mi><mi>a</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math> latus minus. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>h</mi><mo>-</mo><mi>a</mi><mo>,</mo><mfrac><mrow><mi>b</mi><mi>h</mi><mo>-</mo><mi>b</mi><mi>a</mi></mrow><mrow><mi>f</mi></mrow></mfrac></mstyle></math> latus maius.
<lb/>[<emph style="it">tr:
An addition of my own for the second case another way. <lb/>
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be the portion of the first and smaller line. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math> is the portion of the second and greater line. <lb/>
Since: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>a</mi><mo>,</mo><mfrac><mrow><mi>b</mi><mi>a</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math> for the smaller line. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>h</mi><mo>-</mo><mi>a</mi><mo>,</mo><mfrac><mrow><mi>b</mi><mi>h</mi><mo>-</mo><mi>b</mi><mi>a</mi></mrow><mrow><mi>f</mi></mrow></mfrac></mstyle></math> for the greater line.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1650" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> portio secundi lateris et maioris
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the portion of the second and greater line.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1651" xml:space="preserve">
Nota. <lb/>
Alia etiam est <emph style="st">[???]</emph> <emph style="ins ">zetesis supra</emph> ad <lb/>
investigandum secundum portione in <lb/>
primo casu, videlicet ea quæ <lb/>
est primi lateris.
<lb/>[<emph style="it">tr:
Note. <lb/>
The zetesis above is also another for investigating the second portion in the first case,
namely that which is the first line.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f460v" o="460v" n="921"/>
<pb file="add_6782_f461" o="461" n="922"/>
<div xml:id="echoid-div302" type="page_commentary" level="2" n="302">
<p>
<s xml:id="echoid-s1652" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1652" xml:space="preserve">
This page continues Harriot's work on Zetetic 9 from Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book I.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum IX <lb/>
Invenire duo latera, quorum differentia sit ea quæ præscribitur,
&amp; præterea præfinitæ unciæ lateris unius, adjectæ præfinitis unciis alterius,
æquabunt summam præscriptam.
</quote>
<lb/>
<quote>
To find two lines, whose difference is prescribed,
and also such that a fixed part of one line added to a fixed part of the other is equal to a prescibed sum.
</quote>
<lb/>
<s xml:id="echoid-s1653" xml:space="preserve">
Viète used the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the difference between the two lines,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the first part to the first line,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the second part to the second line,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> for the prescribed sum,
and <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> for the parts of the first and second lines. <lb/>
Harriot followed Viète's method on Add MS 6782, f. 462.
Here he works the same problem using numbers instead of lengths.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head213" xml:space="preserve" xml:lang="lat">
Zetet. Lib. 1. Zet. 9.
<lb/>[<emph style="it">tr:
Zetetica, Book I, Zetetic 9
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1655" xml:space="preserve">
Invenire duas numerus quorum differentia sit 84, et præterea <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> <emph style="st">unius</emph> primi <lb/>
numeri adjecta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> alterius æquabit summam præscriptam. oportet summam præscriptam <lb/>
esse maiorem quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> dictæ differentiæ si primum latus sit maius: sed si minus <lb/>
esse maiorem quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>.
<lb/>[<emph style="it">tr:
To find two numbers whose dfference is 84, and also such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> of the first number
added to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of the other will be equal to a prescribed sum;
the prescribed sum must be greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> of the said difference if the first root is the greater,
but if it is the smaller, it must be greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1656" xml:space="preserve">
vel: Invenire duas numerus quorum differentia sit 84, et pæterea <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> primi numeri <lb/>
adjecta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> secunda æquabit summam præscriptam. oportet summam præscriptam <lb/>
esse maiorem quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> dictæ differentiæ si primum latus sit maius: sed si minus <lb/>
esse maiorem quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math>.
<lb/>[<emph style="it">tr:
Or: To find two numbers whose dfference is 84, and also such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of the first number
added to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> of the secon will be equal to a prescribed sum;
the prescribed sum must be greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of the said difference if the first root is the greater,
but if it is the smaller, it must be greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1657" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. differntia numerourm. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. primus numerus <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. secundus numerus. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. portio primi numeri. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. portio 2<emph style="super">i</emph> numeri <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>e</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> summa præscripta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>a</mi><mo>+</mo><mi>e</mi></mstyle></math> <lb/>
Quæruntur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, <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>.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, the difference between the numbers. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, the first number <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, the second number. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, the portion of the first number. <lb/>
The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, the portion of the first number. <lb/>
The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>e</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>, the prescribed sum, equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>+</mo><mi>e</mi></mstyle></math>. <lb/>
There are sought <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, <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>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1658" xml:space="preserve">
Primum latus maius. Portio maiorum uncium.
<lb/>[<emph style="it">tr:
The first root greater. Portion greater than the fraction.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1659" xml:space="preserve">
Primum latus minus. Portio maiorum uncium.
<lb/>[<emph style="it">tr:
The first root smaller. Portion greater than the fraction.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1660" xml:space="preserve">
Primum latus maius. Portio minorum uncium.
<lb/>[<emph style="it">tr:
The first root greater. Portion less than the fraction.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1661" xml:space="preserve">
Primum latus minus. Portio minorum uncium.
<lb/>[<emph style="it">tr:
The first root smaller. Portion less than the fraction.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1662" xml:space="preserve">
Primum latus minus. Ubi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> est maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> et minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
The first root smaller, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> but less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f461v" o="461v" n="923"/>
<pb file="add_6782_f462" o="462" n="924"/>
<div xml:id="echoid-div303" type="page_commentary" level="2" n="303">
<p>
<s xml:id="echoid-s1663" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1663" xml:space="preserve">
This page continues Harriot's working of Zetetic 8 from Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book I.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum VIII <lb/>
Datum latus ita secare, ut præfinitae unciae segmenti, multatæ præfinitis unciis secundi segmenti:
æquent differentiam praescriptam.
</quote>
<lb/>
<quote>
To cut a given line in such a way that a fixed part of one segment, subtracted from a fixed part of the second segment,
is equal to a prescribed difference.
</quote>
<lb/>
<s xml:id="echoid-s1664" xml:space="preserve">
Viète used the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the whole line,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the first part to the first segment,
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the second part to the second segment,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> for the prescribed difference,
and <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> for the unknown parts of the first and second segments.
Harriot followed Viète's method on Add MS 6782, f. 464.
Here he works the same problem using numbers instead of lengths.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head214" xml:space="preserve" xml:lang="lat">
Zetet. Lib. 1. Zet. 8.
<lb/>[<emph style="it">tr:
Zetetica, Book I, Zetetic 8
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1666" xml:space="preserve">
Datum numerum ita dividere in duas partes, ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> primæ partis, minus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> secundæ partis:
sit æqualis numero pæscripto. oportuit numerum præscriptum esse minorum quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> totius.
<lb/>[<emph style="it">tr:
To divide a given number into two parts, so that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> of the first part minus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of the second part
is equal to a prescribed number; the prescribed number must be less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> of the whole.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1667" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. numerus datus. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. prima pars <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. secunda pars. <lb/>
<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><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> primæ partis. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. (minor quam 28) numerus præscriptus. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>h</mi></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>. secundæ partis. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>a</mi><mo>-</mo><mi>h</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math>
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, the given number. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, the first part <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, the second part. <lb/>
<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><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> of the first part. <lb/>
The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> (less than 28) is the prescribed number. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>h</mi></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> of the second part <lb/>
The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>a</mi><mo>-</mo><mi>h</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1668" xml:space="preserve">
SecundÃ²: <lb/>
Ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> primæ partis minus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> secundæ <lb/>
sit æqualis numero præscripto. oportet <lb/>
numerum præscriptum esse minorem quam <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> totius.
<lb/>[<emph style="it">tr:
Second. <lb/>
As <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of the first part minus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> of the second is equal to the prescribed number,
the prescribed number must be less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of the total.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1669" xml:space="preserve">
Analogia solvens problema.
<lb/>[<emph style="it">tr:
A ratio for solving the problem.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1670" xml:space="preserve">
Aliud exemplum. ubi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> sit 24. minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> et maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. <lb/>
<lb/>[<emph style="it">tr:
Another example, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is 24, less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> and greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1671" xml:space="preserve">
Exemplum 2<emph style="super">i</emph> casus. <lb/>
sit iam <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><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>. primæ partis. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>h</mi></mstyle></math> secundæ partis
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>a</mi><mo>-</mo><mi>h</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math>
<lb/>[<emph style="it">tr:
Example 2. <lb/>
Now let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be a <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of the first part. <lb/>
The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>h</mi></mstyle></math> is the second part.
The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>a</mi><mo>-</mo><mi>h</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math>
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f462v" o="462v" n="925"/>
<pb file="add_6782_f463" o="463" n="926"/>
<div xml:id="echoid-div304" type="page_commentary" level="2" n="304">
<p>
<s xml:id="echoid-s1672" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1672" xml:space="preserve">
This page contains Harriot's working of Zetetic 1 from Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book I.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum I <lb/>
Data differentia duorum laterum, &amp; adgregato eorumdem, invenire latera.
</quote>
<lb/>
<quote>
Given the difference of two roots, and their sum, find the roots.
</quote>
<lb/>
<s xml:id="echoid-s1673" xml:space="preserve">
Viète used the letters <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> for the two roots, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for their difference, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> for their sum.
Harriot repeated Viète's working in his own symbolic notation.
In the lower half of the page, he refers to a proposition in Viète's
<emph style="it">Effectionum geometricarum</emph>, Proposition 12,
where a similar problem is solved geometrcially.
In Harriot's diagrams, <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> represent the two extremes.
</s>
<lb/>
<quote xml:lang="lat">
Propositio XII <lb/>
Data media trium proportionalium &amp; differentia extremarum, invenire extremas.
</quote>
<lb/>
<s xml:id="echoid-s1674" xml:space="preserve">
Given the mean of three proportionals, and the difference of their extremes, find the extremes.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head215" xml:space="preserve" xml:lang="lat">
Zet. 1. lib. 1.
<lb/>[<emph style="it">tr:
Zetetic 1, Book I.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1676" xml:space="preserve">
alia diagrapha. <lb/>
consule 12. p. effect.
<lb/>[<emph style="it">tr:
another diagram; see Effectionum, proposition 12.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f463v" o="463v" n="927"/>
<pb file="add_6782_f464" o="464" n="928"/>
<div xml:id="echoid-div305" type="page_commentary" level="2" n="305">
<p>
<s xml:id="echoid-s1677" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1677" xml:space="preserve">
This page contains Harriot's working of Zetetic 8 from Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book I.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum VIII <lb/>
Datum latus ita secare, ut præfinitae unciae segmenti, multatæ præfinitis unciis secundi segmenti:
æquent differentiam praescriptam.
</quote>
<lb/>
<quote>
To cut a given line in such a way that a fixed part of one segment, subtracted from a fixed part of the second segment,
is equal to a prescribed difference.
</quote>
<lb/>
<s xml:id="echoid-s1678" xml:space="preserve">
Viète used the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the whole line,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the first part to the first segment,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the second part to the second segment,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> for the prescribed difference,
and <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> for the parts of the first and second segments.
Harriot repeated Viète's working in his own notation, and also added some variants of his own. <lb/>
The work is continued using numerical examples on Add MS 6782, f. 462.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head216" xml:space="preserve" xml:lang="lat">
Zetet. Lib. 1. Zet. 8.
<lb/>[<emph style="it">tr:
Zetetica, Book I, Zetetic 8
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1680" xml:space="preserve">
Datum latus ita secare, ut præfinitae unciæ segmenti, multatæ præfinitis unciis secundi segmenti:
æquent differentiam præscriptam.
<lb/>[<emph style="it">tr:
To cut a given line in such a way that a fixed part of one segment, subtracted from a fixed part of the second segment,
is equal to a prescribed difference.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1681" xml:space="preserve">
Nota. <lb/>
Differentia præscripta <lb/>
videlicet <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> debet esse <lb/>
minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> <lb/>
sive <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> sit maior vel <lb/>
minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>, ut <lb/>
infra patebit. <lb/>
Hic sequens argu-<lb/>
mentatio <emph style="super">est</emph> firma ad <lb/>
utraque casum.
<lb/>[<emph style="it">tr:
Note. <lb/>
The prescribed difference, namely <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>, must be less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>
whether <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> is greater or less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>, as is shown below.
Here the following argument is sound in either case.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1682" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. latus secandam. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. primum segmentum <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. secundum seg. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. portio primæ seg. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math> <lb/>
Hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>a</mi><mo>,</mo><mi>o</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. differentia præscripta <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>a</mi><mo>-</mo><mi>h</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math> <lb/>
Hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>a</mi><mo>-</mo><mi>h</mi><mo>,</mo><mi>u</mi></mstyle></math> <lb/>
Quæruntur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, et segmentorum portiones. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>a</mi><mo>,</mo><mfrac><mrow><mi>b</mi><mi>a</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math>. primum seg. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>a</mi><mo>-</mo><mi>h</mi><mo>,</mo><mfrac><mrow><mi>b</mi><mi>a</mi><mo>-</mo><mi>b</mi><mi>h</mi></mrow><mrow><mi>f</mi></mrow></mfrac></mstyle></math>. secundum seg. <lb/>
<lb/>[...]<lb/> <lb/>
Notum. Inde primum seg <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>b</mi><mi>a</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math>.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, the line to be cut. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, the first segment <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, the second segment. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, the portion of the first segment. <lb/>
The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math> <lb/>
That is: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>a</mi><mo>:</mo><mi>o</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>, the prescribed difference <lb/>
The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>a</mi><mo>-</mo><mi>h</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math> <lb/>
That is: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>a</mi><mo>-</mo><mi>h</mi><mo>:</mo><mi>u</mi></mstyle></math> <lb/>
There are sought <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>u</mi></mstyle></math>, and the portions of the segments. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>a</mi><mo>:</mo><mfrac><mrow><mi>b</mi><mi>a</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math>, the first segment <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>a</mi><mo>-</mo><mi>h</mi><mo>:</mo><mfrac><mrow><mi>b</mi><mi>a</mi><mo>-</mo><mi>b</mi><mi>h</mi></mrow><mrow><mi>f</mi></mrow></mfrac></mstyle></math>, the second segement. <lb/>
<lb/>[...]<lb/> <lb/>
Note. Thence the first segment, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>b</mi><mi>a</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1683" xml:space="preserve">
PorrÃ², sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> portio <lb/>
secundi segmenti.
<lb/>[<emph style="it">tr:
Further, if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> is the portion of the second segment.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1684" xml:space="preserve">
Additio nostra pro <lb/>
portione secundi segmenti, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> <lb/>
aliter. <lb/>
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, portio secundi segmenti <lb/>
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>+</mo><mi>h</mi></mstyle></math> portio primi segmenti. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>e</mi><mo>,</mo><mfrac><mrow><mi>b</mi><mi>e</mi></mrow><mrow><mi>f</mi></mrow></mfrac></mstyle></math>. secundum seg. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>e</mi><mo>+</mo><mi>h</mi><mo>,</mo><mfrac><mrow><mi>b</mi><mi>e</mi><mo>+</mo><mi>b</mi><mi>h</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math>. prim. seg. <lb/>
<lb/>[...]<lb/> <lb/>
Notum inde sec. segmentum: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>b</mi><mi>e</mi></mrow><mrow><mi>f</mi></mrow></mfrac></mstyle></math>.
<lb/>[<emph style="it">tr:
An addition of my own for the portion of the second segment, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, another way. <lb/>
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the portion of the second segment, therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>+</mo><mi>h</mi></mstyle></math> is the portion of the first segment. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>e</mi><mo>:</mo><mfrac><mrow><mi>b</mi><mi>e</mi></mrow><mrow><mi>f</mi></mrow></mfrac></mstyle></math>, the second segment. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>e</mi><mo>+</mo><mi>h</mi><mo>:</mo><mfrac><mrow><mi>b</mi><mi>e</mi><mo>+</mo><mi>b</mi><mi>h</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math>, the first segment. <lb/>
<lb/>[...]<lb/> <lb/>
Note, thence the second segment: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>b</mi><mi>e</mi></mrow><mrow><mi>f</mi></mrow></mfrac></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1685" xml:space="preserve">
Additio nostra. Aliter 1<emph style="super">o</emph>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>,</mo><mi>d</mi><mo>:</mo><mi>o</mi><mo>,</mo><mfrac><mrow><mi>b</mi><mi>o</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math>. port. 1. seg. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>,</mo><mi>f</mi><mo>:</mo><mi>b</mi><mo>-</mo><mi>o</mi><mo>,</mo><mfrac><mrow><mi>f</mi><mi>b</mi><mo>-</mo><mi>f</mi><mi>o</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>. por. 2. seg.
<lb/>[<emph style="it">tr:
An addition of my own. Another way, 1. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>:</mo><mi>d</mi><mo>=</mo><mi>o</mi><mo>:</mo><mfrac><mrow><mi>b</mi><mi>o</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math>, the portion of the first segment. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>:</mo><mi>f</mi><mo>=</mo><mi>b</mi><mo>-</mo><mi>o</mi><mo>:</mo><mfrac><mrow><mi>f</mi><mi>b</mi><mo>-</mo><mi>f</mi><mi>o</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>, the portion of the second segment.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1686" xml:space="preserve">
Aliter 2<emph style="super">o</emph>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>,</mo><mi>f</mi><mo>:</mo><mi>u</mi><mo>,</mo><mfrac><mrow><mi>f</mi><mi>u</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>. port. 2. seg. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>,</mo><mi>d</mi><mo>:</mo><mi>b</mi><mo>-</mo><mi>u</mi><mo>,</mo><mfrac><mrow><mi>d</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>u</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>. por. 1. seg.
<lb/>[<emph style="it">tr:
Another way, 2. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>:</mo><mi>f</mi><mo>=</mo><mi>u</mi><mo>:</mo><mfrac><mrow><mi>f</mi><mi>u</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>, the portion of the second segment. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>:</mo><mi>d</mi><mo>=</mo><mi>b</mi><mo>-</mo><mi>u</mi><mo>:</mo><mfrac><mrow><mi>d</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>u</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>, the portion of the first segment.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1687" xml:space="preserve">
Nota. Hinc apparet quod <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> est minor <lb/>
quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>; alias <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> esset minor <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>; totum [???] partem.
<lb/>[<emph style="it">tr:
Note. Here it is clear that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>;
otherwise <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> would be less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>; the whole [???] the part.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f464v" o="464v" n="929"/>
<pb file="add_6782_f465" o="465" n="930"/>
<div xml:id="echoid-div306" type="page_commentary" level="2" n="306">
<p>
<s xml:id="echoid-s1688" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1688" xml:space="preserve">
This page contains Harriot's working of Zetetic 6 from Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book I.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum VI <lb/>
Datis duobus lateribs uno deficiente Ã  justo, altero justum excedente, una cum ratione defectus ad excessum:
invenire latus justum.
</quote>
<lb/>
<quote>
Given two roots, one less than the correct root, the other exceeding it,
together with the ratio of the defect to the excess, find the correct root.
</quote>
<lb/>
<s xml:id="echoid-s1689" xml:space="preserve">
Viète used the letters <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>D</mi></mstyle></math> for the two roots, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>R</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>S</mi></mstyle></math> for the ratio of the defect to the excess.
Harriot repeated Viète's working, including his alternative method ('Aliter'), in his own symbolic notation.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head217" xml:space="preserve" xml:lang="lat">
Zetet. lib. 1. Zet. 6.
<lb/>[<emph style="it">tr:
Zetetica, Book I, Zetetic 6.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1691" xml:space="preserve">
Datis duobus lateribs uno deficiente Ã  justo, altero justum excedente, una <lb/>
cum ratione defectus ad excessum: invenire latus justum.
<lb/>[<emph style="it">tr:
Given two roots, one less than the correct side, the other exceeding it,
together with the ratio of the deficiency to the excess, find the correct root.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1692" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. deficiens a justo. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. excedens justium. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>. defectus <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>. excessus ratione
primÃ². <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. esto defectus a justo. <lb/>
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>a</mi></mstyle></math>. latus iustum. <lb/>
Quoniam: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>,</mo><mi>s</mi><mo>:</mo><mi>a</mi><mo>,</mo><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mstyle></math> excessus <lb/>
<lb/>[...]<lb/> <lb/>
Ergo 80 latus iustum.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the deficiency from the correct root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> the excess over the correct root,
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> the ratio of the defect to the excess. <lb/>
First, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be the defect from the correct side. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>a</mi></mstyle></math> is the correct root. <lb/>
Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>:</mo><mi>s</mi><mo>=</mo><mi>a</mi><mo>:</mo><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mstyle></math>, the excess. <lb/>
<lb/>[...]<lb/> <lb/>
Therefore 80 is the correct root.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1693" xml:space="preserve">
SecundÃ² <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. esto excessus. <lb/>
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>e</mi></mstyle></math>. latus iustum. <lb/>
Tum: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mo>,</mo><mi>r</mi><mo>:</mo><mi>e</mi><mo>,</mo><mfrac><mrow><mi>r</mi><mi>e</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mstyle></math>. defectus. <lb/>
<lb/>[...]<lb/> <lb/>
Ergo 80 latus iustum.
<lb/>[<emph style="it">tr:
Second. <lb/>
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the excess. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>e</mi></mstyle></math> is the correct root. <lb/>
Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mo>:</mo><mi>r</mi><mo>=</mo><mi>e</mi><mo>:</mo><mfrac><mrow><mi>r</mi><mi>e</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mstyle></math>, the defect. <lb/>
<lb/>[...]<lb/> <lb/>
Therefore 80 is the correct root.
</emph>]<lb/>
</s>
</p>
<head xml:id="echoid-head218" xml:space="preserve" xml:lang="lat">
Aliter
<lb/>[<emph style="it">tr:
Another way.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1694" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. esto latus iustum. <lb/>
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>b</mi></mstyle></math>. defectus. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>a</mi></mstyle></math>. excessus.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be the correct root. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>a</mi></mstyle></math> is the defect, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>a</mi></mstyle></math> the excess.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f465v" o="465v" n="931"/>
<pb file="add_6782_f466" o="466" n="932"/>
<div xml:id="echoid-div307" type="page_commentary" level="2" n="307">
<p>
<s xml:id="echoid-s1695" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1695" xml:space="preserve">
This page contains Harriot's working of Zetetic 5 from Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book I.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum V <lb/>
Datis duobus lateribus excedentibus justum, una cum ratione excessuum: invenire latus justum.
</quote>
<lb/>
<quote>
Given two roots exceeding the correct one, and the ratio of the excesses, find the correct root.
</quote>
<lb/>
<s xml:id="echoid-s1696" xml:space="preserve">
Viète used the letters <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>D</mi></mstyle></math> for the two roots, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>R</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>S</mi></mstyle></math> for the ratio of the excesses.
Harriot repeated Viète's working, including his alternative method ('Aliter'), in his own symbolic notation.
For Harriot's (and Viète's) use of the symbol that looks like a modern = sign,
see the commentary to Add MS 6782, f. 467.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head219" xml:space="preserve" xml:lang="lat">
Zetet. lib. 1. Zet. 5.
<lb/>[<emph style="it">tr:
Zetetica, Book I, Zetetic 5.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1698" xml:space="preserve">
Datis duobus lateribus excedentibus iustum, una cum ratione excessum: <lb/>
invenire latus iustum.
<lb/>[<emph style="it">tr:
Given two roots exceeding the correct root, and the ratio of the excesses, find the correct root.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1699" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. latus primum. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. latus secundum. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>. excessus primi <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>. excessus secundi ratione
primÃ². <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. esto excessus primi. <lb/>
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>a</mi></mstyle></math>. latus iustum. <lb/>
Tum: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>,</mo><mi>s</mi><mo>:</mo><mi>a</mi><mo>,</mo><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mstyle></math> Excessus 2<emph style="super">i</emph> <lb/>
<lb/>[...]<lb/> <lb/>
Ergo latus iustum 20.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the first side, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> the second, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> the ratio of the excesses. <lb/>
First, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be the first excess. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>a</mi></mstyle></math> is the correct root. <lb/>
Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>:</mo><mi>s</mi><mo>=</mo><mi>a</mi><mo>:</mo><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mstyle></math>, the second excess. <lb/>
<lb/>[...]<lb/> <lb/>
Therefore the correct root is 20.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1700" xml:space="preserve">
SecundÃ² <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. esto excessus secundi. <lb/>
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>e</mi></mstyle></math>. latus iustum. <lb/>
Tum: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mo>,</mo><mi>r</mi><mo>:</mo><mi>e</mi><mo>,</mo><mfrac><mrow><mi>r</mi><mi>e</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mstyle></math>. excessus primi. <lb/>
<lb/>[...]<lb/> <lb/>
Ergo latus iustum 30.
<lb/>[<emph style="it">tr:
Second. <lb/>
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the second excess. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>e</mi></mstyle></math> is the correct root. <lb/>
Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mo>:</mo><mi>r</mi><mo>=</mo><mi>e</mi><mo>:</mo><mfrac><mrow><mi>r</mi><mi>e</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mstyle></math>, the first excess. <lb/>
<lb/>[...]<lb/> <lb/>
Therefore the correct root is 30.
</emph>]<lb/>
</s>
</p>
<head xml:id="echoid-head220" xml:space="preserve" xml:lang="lat">
Aliter
<lb/>[<emph style="it">tr:
Another way.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1701" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. esto latus iustum. <lb/>
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>a</mi></mstyle></math>. excessus primi. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>a</mi></mstyle></math>. excessus secundi.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be the correct root. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>a</mi></mstyle></math> is the first excess, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>a</mi></mstyle></math> the second excess.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1702" xml:space="preserve">
20. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. latus iustum
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn><mn>0</mn></mstyle></math>, the correct root.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1703" xml:space="preserve">
Nota. <lb/>
Si notatio ita fiat. sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. latus minus <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. latus maius <lb/>
&amp;c. <lb/>
Tunc non opus esset isto <lb/>
signo =.
<lb/>[<emph style="it">tr:
Note. <lb/>
Of the notation is thus: let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the smaller side, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> the larger, etc. <lb/>
Then there would be no need for this sign =.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f466v" o="466v" n="933"/>
<pb file="add_6782_f467" o="467" n="934"/>
<div xml:id="echoid-div308" type="page_commentary" level="2" n="308">
<p>
<s xml:id="echoid-s1704" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1704" xml:space="preserve">
This page contains Harriot's working of Zetetic 4 from Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book I.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum IV <lb/>
Datis duobus lateribus deficientibus Ã  justo, una cum ratione defectum: invenire latus justum.
</quote>
<lb/>
<quote>
Given two roots less than the correct one, and the ratio of the defects, find the correct root.
</quote>
<lb/>
<s xml:id="echoid-s1705" xml:space="preserve">
Viète used the letters <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>D</mi></mstyle></math> for the two roots, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>R</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>S</mi></mstyle></math> for the ratio of the defects.
Harriot repeated Viète's working, including his alternative method ('Aliter'),in his own symbolic notation.
The symbol that looks like a modern = sign is to be read as a minus sign,
used by Harriot (following Viète) in cases where it was not known which quantity was greater.
Thus, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math> is to be read as '<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>b</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>a</mi></mstyle></math>, whichever is positive'.
In modern notation the same result is represented by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo lspace="0em" rspace="0em" maxsize="1">|</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo lspace="0em" rspace="0em" maxsize="1">|</mo></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head221" xml:space="preserve" xml:lang="lat">
Zetet. lib. 1. Zet. 4.
<lb/>[<emph style="it">tr:
Zetetica, Book I, Zetetic 4
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1707" xml:space="preserve">
Datis duobus lateribus deficientibus a iusto, una cum ratione defectum: <lb/>
invenire latus iustum.
<lb/>[<emph style="it">tr:
Given two roots less than the true one, together with the ratio of the defects, find the true root.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1708" xml:space="preserve">
primÃ². <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. latus primum. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. latus secundum. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>. defectus primum <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>. defectus secundi ratione
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. esto defectus primi. <lb/>
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>a</mi></mstyle></math>. latus iustum. <lb/>
Tum: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>,</mo><mi>s</mi><mo>:</mo><mi>a</mi><mo>,</mo><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mstyle></math>
defect. 2<emph style="super">i</emph> <lb/>
<lb/>[...]<lb/> <lb/>
Ergo latus iustum: 60.
<lb/>[<emph style="it">tr:
First. <lb/>
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the first root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> the second, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> the ratio of the first defect to the second,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> the first defect. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>a</mi></mstyle></math> is the true root. <lb/>
Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>:</mo><mi>s</mi><mo>=</mo><mi>a</mi><mo>:</mo><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mstyle></math>, the second defect. <lb/>
<lb/>[...]<lb/> <lb/>
Therefore the correct root is 60.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1709" xml:space="preserve">
secundÃ² <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. esto defectus secundi. <lb/>
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>+</mo><mi>e</mi></mstyle></math>. latus iustum. <lb/>
Tum: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mo>,</mo><mi>r</mi><mo>:</mo><mi>e</mi><mo>,</mo><mfrac><mrow><mi>r</mi><mi>e</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mstyle></math>. <lb/>
<lb/>[...]<lb/> <lb/>
Ergo latus iustum: 60.
<lb/>[<emph style="it">tr:
Second <lb/>
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the second defect. <lb/>
Therefore the true root is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>+</mo><mi>e</mi></mstyle></math>. <lb/>
Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mo>:</mo><mi>r</mi><mo>=</mo><mi>e</mi><mo>:</mo><mfrac><mrow><mi>r</mi><mi>e</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mstyle></math>. <lb/>
<lb/>[...]<lb/> <lb/>
Therefore the correct root is 60.
</emph>]<lb/>
</s>
</p>
<head xml:id="echoid-head222" xml:space="preserve" xml:lang="lat">
Aliter
<lb/>[<emph style="it">tr:
Another way.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1710" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. esto latus iustum. <lb/>
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>b</mi></mstyle></math>. defectus primi. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>d</mi></mstyle></math>. defectus secundi. <lb/>
Quære: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>b</mi><mo>,</mo><mi>a</mi><mo>-</mo><mi>d</mi><mo>:</mo><mi>r</mi><mo>,</mo><mi>s</mi></mstyle></math>. <lb/>
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>a</mi><mo>-</mo><mi>r</mi><mi>d</mi><mo>=</mo><mi>s</mi><mi>a</mi><mo>-</mo><mi>s</mi><mi>b</mi></mstyle></math> <lb/>
Hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mi>b</mi><mo>=</mo><mo>=</mo><mi>r</mi><mi>d</mi><mo>=</mo><mi>s</mi><mi>a</mi><mo>=</mo><mo>=</mo><mi>r</mi><mi>a</mi></mstyle></math>. <lb/>
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>s</mi><mi>b</mi><mo>=</mo><mo>=</mo><mi>r</mi><mi>d</mi></mrow><mrow><mi>s</mi><mo>=</mo><mo>=</mo><mi>r</mi></mrow></mfrac><mo>=</mo><mi>a</mi></mstyle></math>
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be the correct root. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>b</mi></mstyle></math> is the first defect, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>d</mi></mstyle></math> the second. <lb/>
Obtain: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>b</mi><mo>:</mo><mi>a</mi><mo>-</mo><mi>d</mi><mo>=</mo><mi>r</mi><mo>:</mo><mi>s</mi></mstyle></math>. <lb/>
Therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo lspace="0em" rspace="0em" maxsize="1">|</mo><mi>r</mi><mi>a</mi><mo>-</mo><mi>r</mi><mi>d</mi><mo lspace="0em" rspace="0em" maxsize="1">|</mo><mo>=</mo><mo lspace="0em" rspace="0em" maxsize="1">|</mo><mi>s</mi><mi>a</mi><mo>-</mo><mi>s</mi><mi>b</mi><mo lspace="0em" rspace="0em" maxsize="1">|</mo></mstyle></math>. <lb/>
That is: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mo lspace="0em" rspace="0em" maxsize="1">|</mo><mi>s</mi><mi>b</mi><mo>-</mo><mi>r</mi><mi>d</mi><mo lspace="0em" rspace="0em" maxsize="1">|</mo></mrow><mrow><mo lspace="0em" rspace="0em" maxsize="1">|</mo><mi>s</mi><mo>-</mo><mi>r</mi><mo lspace="0em" rspace="0em" maxsize="1">|</mo></mrow></mfrac><mo>=</mo><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1711" xml:space="preserve">
<emph style="st">
Emendata Vieta <lb/>
si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> sit minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>, minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>a</mi><mo>-</mo><mi>r</mi><mi>d</mi><mo>=</mo><mi>s</mi><mi>a</mi><mo>-</mo><mi>s</mi><mi>b</mi></mstyle></math>. <lb/>
Tum si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> sit maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>: vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>. <lb/>
Inde: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>a</mi><mo>-</mo><mi>s</mi><mi>a</mi><mo>=</mo><mi>r</mi><mi>d</mi><mo>-</mo><mi>s</mi><mi>b</mi></mstyle></math> <lb/>
Vel: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>e</mi><mo>+</mo><mi>s</mi><mi>e</mi><mo>=</mo><mi>s</mi><mi>g</mi></mstyle></math>. <lb/>
Unde: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mfrac><mrow><mi>r</mi><mi>d</mi><mo>-</mo><mi>s</mi><mi>b</mi></mrow><mrow><mi>r</mi><mo>-</mo><mi>s</mi></mrow></mfrac></mstyle></math>
<lb/>[<emph style="it">tr:
A correction to Viète. <lb/>
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is smaller than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> is smaller than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>a</mi><mo>-</mo><mi>r</mi><mi>d</mi><mo>-</mo><mi>s</mi><mi>a</mi><mo>-</mo><mi>s</mi><mi>b</mi></mstyle></math>. <lb/>
Then if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>,
thence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>a</mi><mo>-</mo><mi>s</mi><mi>a</mi><mo>-</mo><mi>r</mi><mi>d</mi><mo>-</mo><mi>s</mi><mi>b</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>e</mi><mo>+</mo><mi>s</mi><mi>e</mi><mo>=</mo><mi>s</mi><mi>g</mi></mstyle></math>. Whence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mfrac><mrow><mi>r</mi><mi>d</mi><mo>-</mo><mi>s</mi><mi>b</mi></mrow><mrow><mi>r</mi><mo>-</mo><mi>s</mi></mrow></mfrac></mstyle></math>.
</emph>]<lb/>
</emph>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1712" xml:space="preserve">
Nota <lb/>
Si notatis ita fiat: sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. latus minus. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> latus maius. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> defectus minus <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> defectus maius ratione <lb/>
Tunc non opus est isto signo =
<lb/>[<emph style="it">tr:
Note <lb/>
If the notation is thus: let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the smaller root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> the larger root,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> the ratio of the smaller defect to the larger, then there is no need for this sign ==
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f467v" o="467v" n="935"/>
<div xml:id="echoid-div309" type="page_commentary" level="2" n="309">
<p>
<s xml:id="echoid-s1713" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1713" xml:space="preserve">
This page contains some of Harriot's rough working on Zetetic 9 from Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book I. <lb/>
For fuller treatments of the material see Add MS 6782, f. 460 adn f. 461.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head223" xml:space="preserve" xml:lang="lat">
Zet. 9. Cas. 2
<lb/>[<emph style="it">tr:
Zetetic 9, case 2.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1715" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. portio primi lateris et <emph style="st">maioris</emph> minimis. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math>. portio secundi et maioris <lb/>
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, the portion of the first line, the lesser. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math>, the portion of the second and greater line.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1716" xml:space="preserve">
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. portio secundi lateris et maioris.
<lb/>[<emph style="it">tr:
et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the portion of the second and greater line.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1717" xml:space="preserve">
Primum latus minus. Portio minorum uncium.
<lb/>[<emph style="it">tr:
The first line smaller. Portion less than the fraction.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1718" xml:space="preserve">
Primum latus maius. Portio minorum uncium.
<lb/>[<emph style="it">tr:
The first line greater. Portion less than the fraction.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1719" xml:space="preserve">
Primum latus minus.
<lb/>[<emph style="it">tr:
The first line smaller.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f468" o="468" n="936"/>
<div xml:id="echoid-div310" type="page_commentary" level="2" n="310">
<p>
<s xml:id="echoid-s1720" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1720" xml:space="preserve">
The reference here is to Viète,
<emph style="it">Ad problema, quod ... proposuit Adrianus Romanus, responsum</emph> (1595),
pages 5, 7, and 13.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head224" xml:space="preserve" xml:lang="lat">
ad pag: 5.7. &amp; 13. Vietæ ad Adrianum. videlicet responsi
<lb/>[<emph style="it">tr:
On pages 5, 7, and 13 of Viète, Adrianus, that is, his responses.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1722" xml:space="preserve">
Aliter.
<lb/>[<emph style="it">tr:
Another way.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1723" xml:space="preserve">
Excedit semicirculum
<lb/>[<emph style="it">tr:
Exceeds a semicircle
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f468v" o="468v" n="937"/>
<pb file="add_6782_f469" o="469" n="938"/>
<div xml:id="echoid-div311" type="page_commentary" level="2" n="311">
<p>
<s xml:id="echoid-s1724" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1724" xml:space="preserve">
The positions of this page amongst other pages referring to Viète's
<emph style="it">Adrianus Romanus responsum</emph>, and its subject matter,
trisection and quinquisection of an angle, suggests that it relates to
<emph style="it">Adrianus Romanus responsum</emph>, Chapter V.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s1726" xml:space="preserve">
Pentachotomia
<lb/>[<emph style="it">tr:
Quinquisection of an angle
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1727" xml:space="preserve">
Trichotomia
<lb/>[<emph style="it">tr:
Trisection of an angle
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f469v" o="469v" n="939"/>
<div xml:id="echoid-div312" type="page_commentary" level="2" n="312">
<p>
<s xml:id="echoid-s1728" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1728" xml:space="preserve">
The positions of this page amongst other pages referring to Viète's
<emph style="it">Adrianus Romanus responsum</emph>, and its subject matter,
trisection and quinquisection of an angle, suggests that it relates to
<emph style="it">Adrianus Romanus responsum</emph>, Chapter V.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s1730" xml:space="preserve">
Pentachotomia
<lb/>[<emph style="it">tr:
Quinquisection of an angle
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f470" o="470" n="940"/>
<div xml:id="echoid-div313" type="page_commentary" level="2" n="313">
<p>
<s xml:id="echoid-s1731" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1731" xml:space="preserve">
The references on this page are to Viète,
<emph style="it">Ad problema, quod ... proposuit Adrianus Romanus, responsum</emph> (1595),
pages 7v and 5.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head225" xml:space="preserve" xml:lang="lat">
ad: pag: 7.b.
<lb/>[<emph style="it">tr:
On page 7v.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1733" xml:space="preserve">
omisit Vieta radices negativas <lb/>
ut in omnibus alijs aequationibus
<lb/>[<emph style="it">tr:
Viète omits negative roots, as in all other equations.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1734" xml:space="preserve">
Exemplum pag: 5. pono sub meliori forma <lb/>
quam Vieta, ut sequitur: <lb/>
et est generalis ad omnes sectiones anguli <lb/>
imparis numeri.
<lb/>[<emph style="it">tr:
Example from page 5, I put in a better form than Viète, as follows;
and it is general for all angular sections of odd number.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f470v" o="470v" n="941"/>
<div xml:id="echoid-div314" type="page_commentary" level="2" n="314">
<p>
<s xml:id="echoid-s1735" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1735" xml:space="preserve">
A continuation from Add MS 6782, f. 470, of work on Viète's,
<emph style="it">Ad problema, quod ... proposuit Adrianus Romanus, responsum</emph> (1595).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s1737" xml:space="preserve">
aliter: sed melius in alia charta
<lb/>[<emph style="it">tr:
another way, but better in the other sheet
</emph>]<lb/>
[<emph style="it">Note:
The other sheet referred to here is the reverse of this one, Add MS 6782, f. 470.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1738" xml:space="preserve">
Trichotomia. sic brevie
<lb/>[<emph style="it">tr:
Trisection, thus briefly
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1739" xml:space="preserve">
hinc sequitur:
<lb/>[<emph style="it">tr:
this follows:
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f471" o="471" n="942"/>
<div xml:id="echoid-div315" type="page_commentary" level="2" n="315">
<p>
<s xml:id="echoid-s1740" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1740" xml:space="preserve">
The positions of this page amongst other pages referring to Viète's
<emph style="it">Adrianus Romanus responsum</emph>, and its subject matter,
trisection of an angle, suggests that it relates to
<emph style="it">Adrianus Romanus responsum</emph>, Chapter V.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head226" xml:space="preserve" xml:lang="lat">
Trichotomia
<lb/>[<emph style="it">tr:
Trisection
</emph>]<lb/>
</head>
<pb file="add_6782_f471v" o="471v" n="943"/>
<pb file="add_6782_f472" o="472" n="944"/>
<div xml:id="echoid-div316" type="page_commentary" level="2" n="316">
<p>
<s xml:id="echoid-s1742" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1742" xml:space="preserve">
This page contains further work on Zetetic 2 from Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book II.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum II <lb/>
Dato rectangulo sub lateribus, &amp; adgregato quadratorum, inveniuntur latera.
</quote>
<lb/>
<quote>
Given the product of two sides, and the sum of their squares, the sides may be found.
</quote>
<lb/>
<s xml:id="echoid-s1743" xml:space="preserve">
Harriot began the problem on Add MS 6782, f. 472, but he described the treatment on this page as better.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head227" xml:space="preserve" xml:lang="lat">
lib. 2. Zetet. Zet. 2. secundo et melius.
<lb/>[<emph style="it">tr:
Zetetica, Book II, Zetetic 2. Second way, and better.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1745" xml:space="preserve">
Dato rectangulo sub lateribus et adgregato quadratorum, inveniuntur latera. <lb/>
Enimvero duplum planum sub lateribus adiectum quidem <lb/>
adgregatum quadratorum, æquatur quadrato summa laterum:] <lb/>
sit planum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>δ</mi></mstyle></math>, cui æquale <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>θ</mi></mstyle></math>. et adgregatum <lb/>
quadratorum laterum sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>γ</mi><mi>H</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>λ</mi></mstyle></math>. <lb/>
<lb/>[...]<lb/> <lb/>
Ergo datur summa latera.
<lb/>[<emph style="it">tr:
Given a rectangle from its sides, and the sum of their squares, there may be found the sides. <lb/>
Certainly, twic the rectangle from the sides added to the sum of the squares
is equal to the square of the sum of the sides. <lb/>
Let the plane be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>δ</mi></mstyle></math>, which is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>θ</mi></mstyle></math>,
and the sum of the squares of the sides is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>γ</mi><mi>H</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>λ</mi></mstyle></math>. <lb/>
<lb/>[...]<lb/> <lb/>
Therefore the sum of the sides is given.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1746" xml:space="preserve">
Ablatum uno, quadrato differentiæ.] <lb/>
<lb/>[...]<lb/> <lb/>
ergo, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>\</mo><mi>a</mi><mi>l</mi><mi>p</mi><mi>h</mi><mi>μ</mi></mstyle></math> est differentia laterum <lb/>
et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ρ</mi><mi>o</mi></mstyle></math> est quadratum differenti
<lb/>[<emph style="it">tr:
Subtracting from one, the square of the difference, <lb/>
<lb/>[...]<lb/>
therefore, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>μ</mi></mstyle></math> is the difference of the sides, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ρ</mi><mi>o</mi></mstyle></math> is the square of the difference.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1747" xml:space="preserve">
In notis: sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>δ</mi></mstyle></math> planum æquale <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math> . <lb/>
Ergo, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>+</mo><mn>2</mn><mo>,</mo><mi>c</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>a</mi></mstyle></math>. adgregati <lb/>
Et. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>-</mo><mn>2</mn><mo>,</mo><mi>c</mi><mi>c</mi><mo>=</mo><mi>e</mi><mi>e</mi></mstyle></math>. differentia <lb/>
Ergo per 1.z et primi lib. dantur latera <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>β</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>δ</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
In letters, let the plane <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>δ</mi></mstyle></math> be equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math>. <lb/>
Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>+</mo><mn>2</mn><mo>,</mo><mi>c</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>a</mi></mstyle></math>, the sum. <lb/>
And <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>-</mo><mn>2</mn><mo>,</mo><mi>c</mi><mi>c</mi><mo>=</mo><mi>e</mi><mi>e</mi></mstyle></math>, the difference <lb/>
Therefore, by Zetetic 1 from the first book, there are given the sides <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>β</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>δ</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1748" xml:space="preserve">
Aliter. <lb/>
Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>x</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. et sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>δ</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Another way. <lb/>
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>x</mi></mstyle></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>δ</mi></mstyle></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f472v" o="472v" n="945"/>
<pb file="add_6782_f473" o="473" n="946"/>
<div xml:id="echoid-div317" type="page_commentary" level="2" n="317">
<p>
<s xml:id="echoid-s1749" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1749" xml:space="preserve">
This page contains some very brief observations on Zetetic 3 from Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book II.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum III <lb/>
Dato rectangulo sub lateribus, &amp; differentia laterum: inveniuntur latera.
</quote>
<lb/>
<quote>
Given the product of two sides and their difference, the sides may be found.
</quote>
<lb/>
<s xml:id="echoid-s1750" xml:space="preserve">
This is Proposition I.30 from the <emph style="it">Arithmetica</emph> of Diophantus,
but Harriot refers only to Viète's version of it.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head228" xml:space="preserve" xml:lang="lat">
lib. 2. Zet. <lb/>
Zet. 3.
<lb/>[<emph style="it">tr:
Zetetica, Book II, Zetetic 3.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1752" xml:space="preserve">
Dato rectangulo sub lateribus, et differentia laterum inveniuntur latera.
<lb/>[<emph style="it">tr:
Given the product of two sides and their difference, the sides may be found.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1753" xml:space="preserve">
Enimvero:
<lb/>[<emph style="it">tr:
Certainly:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1754" xml:space="preserve">
Sint notæ et in 2, zetetico.
<lb/>[<emph style="it">tr:
This is noted also in Zetetic 2.
</emph>]<lb/>
[<emph style="it">Note:
For Harriot's treatment of Zetetic 2, see Add MS 6782, f. 474 and f. 472.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6782_f473v" o="473v" n="947"/>
<pb file="add_6782_f474" o="474" n="948"/>
<div xml:id="echoid-div318" type="page_commentary" level="2" n="318">
<p>
<s xml:id="echoid-s1755" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1755" xml:space="preserve">
This page contains Harriot's working of Zetetic 2 from Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book II.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum II <lb/>
Dato rectangulo sub lateribus, &amp; adgregato quadratorum, inveniuntur latera.
</quote>
<lb/>
<quote>
Given the product of two sides, and the sum of their squares, the sides may be found.
</quote>
<lb/>
<s xml:id="echoid-s1756" xml:space="preserve">
Harriot's treatment of the problem is continued on Add MS 6782, f. 472.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head229" xml:space="preserve" xml:lang="lat">
lib. 2. Zeteticorum <lb/>
Zet. 2. primo
<lb/>[<emph style="it">tr:
Zetetica, Book II <lb/>
Zetetic 2. First way.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1758" xml:space="preserve">
Dato rectangulo sub lateribus et adgregato quadratorum, inveniuntur latera. <lb/>
Hoc est: <lb/>
Data recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>γ</mi></mstyle></math> et plano <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>δ</mi></mstyle></math>; invenientur <lb/>
latera <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>δ</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>γ</mi></mstyle></math>. <lb/>
vel: <lb/>
Data recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>δ</mi></mstyle></math> et quadrato <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ε</mi></mstyle></math>: invenientur <lb/>
duæ <emph style="super">rectæ</emph> lineæ quæ cum data <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>γ</mi></mstyle></math>,
efficiunt triangulum rectangulum æquale dimidio <lb/>
dati quadrati <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ε</mi></mstyle></math>, ita ut linea data sit Hyponetusu. <lb/>
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>δ</mi></mstyle></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>: Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>γ</mi><mi>δ</mi></mstyle></math> erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>c</mi><mi>c</mi></mrow><mrow><mi>a</mi></mrow></mfrac></mstyle></math>.
<lb/>[<emph style="it">tr:
Given a rectangle from its sides, and the sum of their squares, there may be found the sides. <lb/>
That is: <lb/>
Given the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>γ</mi></mstyle></math> and the surface <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>δ</mi></mstyle></math>,
there may be found the sides <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>δ</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>γ</mi></mstyle></math>. <lb/>
or: <lb/>
Given the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>γ</mi></mstyle></math> and the square <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ε</mi></mstyle></math>,
there may be found two straight lines which, with the given line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>γ</mi></mstyle></math>,
form a right-angled triangle equal to half the given square <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ε</mi></mstyle></math>,
in such a way that the given line is the hypotenuse.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1759" xml:space="preserve">
Aliter. <lb/>
Sit rectangulum datum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>δ</mi></mstyle></math>. cui æquale <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>θ</mi></mstyle></math>. <lb/>
et adgregatum quadratorum sit: <lb/>
<lb/>[...]<lb/> <lb/>
data igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>x</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Another way. <lb/>
Let the given rectangle be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>δ</mi></mstyle></math>, which is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>θ</mi></mstyle></math>,
and the sum of the squares is: <lb/>
<lb/>[...]<lb/> <lb/>
therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>x</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1760" xml:space="preserve">
Itaque in notis: sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>x</mi></mstyle></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> . <lb/>
<lb/>[...]<lb/> <lb/>
Inveniatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, et dicatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Thus in symbols: let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>x</mi></mstyle></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/>
<lb/>[...]<lb/> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> is found, and is said to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1761" xml:space="preserve">
Et <emph style="super">quia</emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, est minor <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
And because <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>,
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f474v" o="474v" n="949"/>
<pb file="add_6782_f475" o="475" n="950"/>
<div xml:id="echoid-div319" type="page_commentary" level="2" n="319">
<p>
<s xml:id="echoid-s1762" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1762" xml:space="preserve">
The reference at the top of this page is to Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book V, Zetetic 14.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum XIV <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> quadratum minus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> plano adaequare uni quadrato, quod fit minus quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>,
sed majus quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>.
</quote>
<lb/>
<quote>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>-squared minus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> is equal to a square, which is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>A</mi></mstyle></math> but greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>A</mi></mstyle></math>.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head230" xml:space="preserve" xml:lang="lat">
Zeteticorum. lib. 5. Zet: 14.
<lb/>[<emph style="it">tr:
Zetetica, Book V, Zetetic 14.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1764" xml:space="preserve">
minus quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>a</mi></mstyle></math>
<lb/>[<emph style="it">tr:
less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>a</mi></mstyle></math>
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1765" xml:space="preserve">
maius quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math>
<lb/>[<emph style="it">tr:
greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math>
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1766" xml:space="preserve">
Quæritur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, ponatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> is sought, suppose it is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1767" xml:space="preserve">
solutio problematis
<lb/>[<emph style="it">tr:
solution to the problem
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f475v" o="475v" n="951"/>
<pb file="add_6782_f476" o="476" n="952"/>
<div xml:id="echoid-div320" type="page_commentary" level="2" n="320">
<p>
<s xml:id="echoid-s1768" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1768" xml:space="preserve">
The reference at the top of this page appears to be to Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book V, Zetetic 14,
as on Add MS 6782, f. 475.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head231" xml:space="preserve" xml:lang="lat">
Zet. lib. Zet: 14.
<lb/>[<emph style="it">tr:
Zetetica, Book [V], Zetetic 14.
</emph>]<lb/>
</head>
<pb file="add_6782_f476v" o="476v" n="953"/>
<pb file="add_6782_f477" o="477" n="954"/>
<div xml:id="echoid-div321" type="page_commentary" level="2" n="321">
<p>
<s xml:id="echoid-s1770" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1770" xml:space="preserve">
This page contains a lemma needed for Zetetic 10 from Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book II.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum X <lb/>
Dato plano, quod constat tum rectangulo sub lateribus, tum quadratis singulorum laterum,
datoque è lateribus uno, invenire latus reliquum.</quote>
<lb/>
<quote>
Given a plane, consisting of a rectangle from two sides together with the individual squares of the sides,
and given also one of the sides, find the remaining side.
</quote>
<lb/>
<s xml:id="echoid-s1771" xml:space="preserve">
Viète called the given side <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>, and the side to be found <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>D</mi></mstyle></math>,
and arrived at the following conclusion:
</s>
<lb/>
<quote xml:lang="">
Planum constans rectangulo sub lateribus, &amp; quadratis singulorum laterum,
multatum dodrante quadrati lateris dati, æquale est quadrato lateris compositi, ex quæsito latere &amp; dimido dati.
</quote>
<lb/>
<quote>
The plane consisting of a rectangle from the sides and the square of the individual sides,
minus three-quarters of the square of the given root,
is equal to the square of the line composed of the sought line and half the given line.
</quote>
<lb/>
<s xml:id="echoid-s1772" xml:space="preserve">
Borrowing Viète's notation, his statement may be written as the identity
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>A</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>D</mi><mrow><msup><mo maxsize="1">)</mo><mn>2</mn></msup></mrow><mo>+</mo><mrow><msup><mi>D</mi><mn>2</mn></msup></mrow><mo>+</mo><mo maxsize="1">(</mo><mi>D</mi><mi>A</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mrow><msup><mi>D</mi><mn>2</mn></msup></mrow><mo maxsize="1">)</mo><mo>=</mo><mrow><msup><mi>A</mi><mn>2</mn></msup></mrow></mstyle></math>.
On this page Harriot investigated the problem geometrically.
On Add MS 6782, f. 479v, he treated it algebraically.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head232" xml:space="preserve" xml:lang="lat">
Lemma. ad Zet. 2, 10.
<lb/>[<emph style="it">tr:
Lemma to Zetetic II.10
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1774" xml:space="preserve">
Si recta linea <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>β</mi></mstyle></math> dividatur bisariam in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>γ</mi></mstyle></math>: et ei adjiciatur altera <lb/>
linea <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>δ</mi></mstyle></math>: dico quod:
<lb/>[<emph style="it">tr:
If the straight line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>β</mi></mstyle></math> is divided in two at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>γ</mi></mstyle></math>,
and to it is added another line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>δ</mi></mstyle></math>, I say that:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1775" xml:space="preserve">
Aliter. <lb/>
Sit rectangulum datum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>δ</mi></mstyle></math>. cui æquale <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>θ</mi></mstyle></math>. <lb/>
et adgregatum quadratorum sit: <lb/>
<lb/>[...]<lb/> <lb/>
data igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>x</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Another way. <lb/>
Let the given rectangle be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>δ</mi></mstyle></math>, which is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>θ</mi></mstyle></math>,
and the sum of the squares is: <lb/>
<lb/>[...]<lb/> <lb/>
therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>x</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f477v" o="477v" n="955"/>
<pb file="add_6782_f478" o="478" n="956"/>
<pb file="add_6782_f478v" o="478v" n="957"/>
<pb file="add_6782_f479" o="479" n="958"/>
<pb file="add_6782_f479v" o="479v" n="959"/>
<div xml:id="echoid-div322" type="page_commentary" level="2" n="322">
<p>
<s xml:id="echoid-s1776" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1776" xml:space="preserve">
There are margin notes on this page referring to Zetetica 9 and 10 from Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book II.
</s>
<lb/>
<quote xml:lang="">
Zeteticum IX <lb/>
Dato rectangulo sub lateribus, &amp; differentia quadratorum, invenire latera.
</quote>
<lb/>
<quote>
Given a rectangle from two sides and the difference of their squares, find the sides.
</quote>
<lb/>
<quote xml:lang="lat">
Zeteticum X <lb/>
Dato plano, quod constat tum rectangulo sub lateribus, tum quadratis singulorum laterum,
datoque è lateribus uno, invenire latus reliquum.</quote>
<lb/>
<quote>
Given a plane, consisting of a rectangle from two sides together with the individual squares of the sides,
and given also one of the sides, find the remaining side.
</quote>
<lb/>
<s xml:id="echoid-s1777" xml:space="preserve">
For a fuller investigation of Zetetic X, see Add MS 6782, f. 477.
</s>
<lb/>
<s xml:id="echoid-s1778" xml:space="preserve">
At the bottom of the page there is what appears to be a list of books.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s1780" xml:space="preserve">
Zet.2,9
<lb/>[<emph style="it">tr:
Zetetic II.9
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1781" xml:space="preserve">
cubuc rectanguli sub <lb/>
lateribus
<lb/>[<emph style="it">tr:
cube of the rectangle from the sides
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1782" xml:space="preserve">
Zet.2,10
<lb/>[<emph style="it">tr:
Zetetic II.10
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s1783" xml:space="preserve">
[???] <lb/>
[???] <lb/>
[???] <lb/>
[???] <lb/>
A bagg of books [???] <lb/>
Vitello <lb/>
Euclid. 2. vol. <lb/>
physica Arist. [???] <lb/>
Ethica Piccolomini
</s>
</p>
<pb file="add_6782_f480" o="480" n="960"/>
<div xml:id="echoid-div323" type="page_commentary" level="2" n="323">
<p>
<s xml:id="echoid-s1784" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1784" xml:space="preserve">
The reference at the top of this page is to Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book V, Zetetic 4.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum IV <lb/>
Invenire numero tria plana, quae bina juncta, ac etiam ipsa trium summa adscito dato plano,
quadratum constituant.
</quote>
<lb/>
<quote>
To find three plane numbers, of which the sum of any two, as also the sum of all three,
added to a given plane, constitutes a square.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head233" xml:space="preserve" xml:lang="lat">
Zet. 4: lib. 5.
<lb/>[<emph style="it">tr:
Zetetic 4, Book V.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1786" xml:space="preserve">
Invenire numero tria plana quæ bina juncta ac etiam ipsa trium <lb/>
summa adscito dato plano Quadratum constituant.
<lb/>[<emph style="it">tr:
To find three plane numbers, of which the sum of any two, as also the sum of all three,
added to a given plane, constitutes a square.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1787" xml:space="preserve">
Sit datum planum.
<lb/>[<emph style="it">tr:
Let the given plane be
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1788" xml:space="preserve">
Aggregatum primi quæsiti <lb/>
plani et secundi
<lb/>[<emph style="it">tr:
The sum of the first plane sought and the second
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1789" xml:space="preserve">
Aggregatum secundi <lb/>
et tertij
<lb/>[<emph style="it">tr:
The sum of the second and the third
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1790" xml:space="preserve">
Summa trium
<lb/>[<emph style="it">tr:
The sum of the three
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1791" xml:space="preserve">
Inde: summa <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo></mstyle></math> aggregato <lb/>
primi et secundi <lb/>
erit tertium planum
<lb/>[<emph style="it">tr:
Whence: the sum, minus the sum of the first and the second, will be the first plane.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1792" xml:space="preserve">
etiam: summa <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo></mstyle></math> aggregato <lb/>
secundi et tertij <lb/>
erit primum planum <lb/>[<emph style="it">tr:
also: the sum, minus the sum of the second and third, will be the first plane
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1793" xml:space="preserve">
Inde: Aggregatum primi et <lb/>
tertij adscito plani <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mn>1</mn></mstyle></math> <lb/>
æquale <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>f</mi></mstyle></math> quadrato
<lb/>[<emph style="it">tr:
Whence: the sum of the first and third added to the plane <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mn>1</mn></mstyle></math> will equal the square <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>f</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f480v" o="480v" n="961"/>
<pb file="add_6782_f481" o="481" n="962"/>
<div xml:id="echoid-div324" type="page_commentary" level="2" n="324">
<p>
<s xml:id="echoid-s1794" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1794" xml:space="preserve">
This page is one of the few that contains a date: 29 August 1600.
The date is consistent with the suggestion that Harriot's friend Nathaniel Torporley met Viète in person
in Paris in the late 1590s, and brought Viète's books back to England (see Stedall 2003). <lb/>
This appears to be the first of several pages in which Harriot worked systematically through Viète's
<emph style="it">Zeteticorum libri quinque</emph> (1591 or 1593),
re-writing the propositions and proofs in his own notation.
He began here with Zetetics 2 and 3 from Book 1; Zetetic 1 is sketched in outline on Add MS 6782, f. 463.
He reached the end of Book I nine days later on 6 September (see Add MS 6782, f. 456.)
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum II <lb/>
Data differentia duorum laterum, &amp; ratione eorumdem, invenire latera.
</quote>
<lb/>
<quote>
Given the difference of two roots, and their ratio, find the roots.
</quote>
<lb/>
<quote xml:lang="lat">
Zeteticum III <lb/>
Data summa laterum, &amp; ratione eorumdem: invenire latera.
</quote>
<lb/>
<quote>
Given the sum of two sides, and their ratio, find the sides.
</quote>
<lb/>
<s xml:id="echoid-s1795" xml:space="preserve">
Viète used the letters <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> for the two roots, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>R</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>S</mi></mstyle></math> for their ratio.
Harriot repeated Viète's working in his own symbolic notation.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head234" xml:space="preserve" xml:lang="lat">
Zetet. lib. 1. Zet. 2. 1600. August. 29.
<lb/>[<emph style="it">tr:
Zetetica, Book I, Zetetic 2. 1600, August 29.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1797" xml:space="preserve">
Datur differentia duorum laterum et ratione eorundem invenire latera.
<lb/>[<emph style="it">tr:
Given the difference of two roots and their ratio, find the roots.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1798" xml:space="preserve">
PrimÃ². <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> differentia <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> minus latus <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> maius latus <emph style="st">in</emph> ratione
latus minus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/>
ergo latus maius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>+</mo><mi>b</mi></mstyle></math>. <lb/>
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>,</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>:</mo><mi>r</mi><mo>,</mo><mi>s</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
First. <lb/>
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the difference, <lb/>
and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> be the ratio of the smaller root to the larger root. <lb/>
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be the smaller root, therefore the larger root is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>+</mo><mi>b</mi></mstyle></math>. <lb/>
Therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>:</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mi>r</mi><mo>:</mo><mi>s</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1799" xml:space="preserve">
SecundÃ² <lb/>
latus maius. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. <lb/>
Ergo latus minus. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>-</mo><mi>b</mi></mstyle></math>. <lb/>
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>,</mo><mi>e</mi><mo>-</mo><mi>b</mi><mo>:</mo><mi>s</mi><mo>,</mo><mi>r</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Second. <lb/>
Let the larger root be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, therefore the smaller root is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>-</mo><mi>b</mi></mstyle></math>. <lb/>
Therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>:</mo><mi>e</mi><mo>-</mo><mi>b</mi><mo>=</mo><mi>s</mi><mo>:</mo><mi>r</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<head xml:id="echoid-head235" xml:space="preserve" xml:lang="lat">
Zet. 3. Data summa laterum, et ratione eorundem invenire latera.
<lb/>[<emph style="it">tr:
Zetetic 3. Given the sum of the roots and their ratio, find the roots.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1800" xml:space="preserve">
primÃ². <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. summa laterum <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>. minus latus <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>. maius latus <emph style="st">in</emph> ratione <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. esto maius latus
ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mo>-</mo><mi>a</mi></mstyle></math> latus minus. <lb/>
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>,</mo><mi>g</mi><mo>-</mo><mi>a</mi><mo>:</mo><mi>r</mi><mo>,</mo><mi>s</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
First. <lb/>
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> be the sum of the roots, <lb/>
and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> be the ratio of the smaller root to the larger root. <lb/>
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be the larger root, therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mo>-</mo><mi>a</mi></mstyle></math> is the smaller root. <lb/>
Therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>:</mo><mi>g</mi><mo>-</mo><mi>a</mi><mo>=</mo><mi>r</mi><mo>:</mo><mi>s</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1801" xml:space="preserve">
secundÃ² <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. esto minus latus. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>-</mo><mi>g</mi></mstyle></math>. maius latus. <lb/>
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>,</mo><mi>g</mi><mo>-</mo><mi>e</mi><mo>:</mo><mi>s</mi><mo>,</mo><mi>r</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Second. <lb/>
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the smaller root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>-</mo><mi>g</mi></mstyle></math> the larger root. <lb/>
Therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>:</mo><mi>g</mi><mo>-</mo><mi>e</mi><mo>=</mo><mi>s</mi><mo>:</mo><mi>r</mi></mstyle></math>.
</emph>]<lb/>
[<emph style="it">Note:
Here <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>-</mo><mi>g</mi></mstyle></math> in the third line is clearly a writing or copying error for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mo>-</mo><mi>e</mi></mstyle></math>;
Harriot has proceeded correctly with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mo>-</mo><mi>e</mi></mstyle></math> in the fourth line.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6782_f481v" o="481v" n="963"/>
<pb file="add_6782_f482" o="482" n="964"/>
<pb file="add_6782_f482v" o="482v" n="965"/>
<pb file="add_6782_f483" o="483" n="966"/>
<pb file="add_6782_f483v" o="483v" n="967"/>
<pb file="add_6782_f484" o="484" n="968"/>
<pb file="add_6782_f484v" o="484v" n="969"/>
<pb file="add_6782_f485" o="485" n="970"/>
<pb file="add_6782_f485v" o="485v" n="971"/>
<pb file="add_6782_f486" o="486" n="972"/>
<pb file="add_6782_f486v" o="486v" n="973"/>
<pb file="add_6782_f487" o="487" n="974"/>
<pb file="add_6782_f487v" o="487v" n="975"/>
<pb file="add_6782_f488" o="488" n="976"/>
<pb file="add_6782_f488v" o="488v" n="977"/>
<pb file="add_6782_f489" o="489" n="978"/>
<pb file="add_6782_f489v" o="489v" n="979"/>
<pb file="add_6782_f490" o="490" n="980"/>
<div xml:id="echoid-div325" type="page_commentary" level="2" n="325">
<p>
<s xml:id="echoid-s1802" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1802" xml:space="preserve">
This page contains work on one of the ratios given by Viète under the heading 'Syntomon'
in Chapter XIX, Proposition 21 of <emph style="it">Variorum responsorum liber VIII</emph>.
For other pages based on the same (36, 45, 70)-degree spherical triangle see Add MS 6787, f. 36 to 39.
</s>
<lb/>
<s xml:id="echoid-s1803" xml:space="preserve">
The reference to Regiomontanus is to his <emph style="it">De triangulis omnimodis</emph>,
Book 5, Proposition 2. For the same proposition, see also Add MS 6787, f. 51.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head236" xml:space="preserve" xml:lang="lat">
Investigatio analogiæ <lb/>
Vietanæ
<lb/>[<emph style="it">tr:
An investigation of one of Viète's ratios
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1805" xml:space="preserve">
(Investigavit Vieta <emph style="super">(ut putamus)</emph> per <lb/>
diagramma Regiomontani <lb/>
lib. 5. pr. de triangulis <lb/>
cum quibusdam additamentis,
ut nos <emph style="st">alibi</emph> etiam alibi.)
<lb/>[<emph style="it">tr:
Viète investigated this (as I believe) from a diagram of Regiomontanus in Book 5 of
<emph style="it">De triangulis</emph> with certain additions, as I have also done elsewhere.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1806" xml:space="preserve">
ut Vieta <lb/>
pag: 47.b.
<lb/>[<emph style="it">tr:
As Viète, page 47v.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1807" xml:space="preserve">
ut Vieta <lb/>
pag: 35.b.
<lb/>[<emph style="it">tr:
As Viète, page 35v.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1808" xml:space="preserve">
conditiones <lb/>
alteri Δ<emph style="super">i</emph> <lb/>
contrariæ
<lb/>[<emph style="it">tr:
conditions in other, opposite triangles
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1809" xml:space="preserve">
Nota. Et si signum (&lt;) ponatur sub <emph style="st">ab</emph>, intelligetur quod ab &lt; 90. <lb/>
Ita signum (&gt;) sub <emph style="st">d</emph>, denotat d &gt; 90. <lb/>
Istud signum, (&lt; &gt;), denotat unum latus maius alterum minus 90. &amp;c.
<lb/>[<emph style="it">tr:
Note. If this sign (&lt;) is placed under <emph style="st">ab</emph>, it is to be understood that ab &lt; 90. <lb/>
This sign (&gt;) under <emph style="st">d</emph>, indicates that d &gt; 90. <lb/>
These signs, (&lt; &gt;), indicate that one side is greater than, the other less than 90. &amp;c.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f490v" o="490v" n="981"/>
<pb file="add_6782_f491" o="491" n="982"/>
<div xml:id="echoid-div326" type="page_commentary" level="2" n="326">
<p>
<s xml:id="echoid-s1810" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1810" xml:space="preserve">
This page contains work on one of the ratios given by Viète under the heading 'Syntomon'
in Chapter XIX, Proposition 21 of <emph style="it">Variorum responsorum liber VIII</emph>.
For other pages based on the same (36, 45, 70)-degree spherical triangle see Add MS 6787, f. 36 to 39.
</s>
<lb/>
<s xml:id="echoid-s1811" xml:space="preserve">
The reference to Regiomontanus is to his <emph style="it">De triangulis omnimodis</emph>,
Book 5, Proposition 2. For the same proposition, see also Add MS 6787, f. 51.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head237" xml:space="preserve" xml:lang="lat">
Investigatio omnium <lb/>
analogiarum et casuum.
<lb/>[<emph style="it">tr:
An investigation of all the ratios and cases.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1813" xml:space="preserve">
Æquationes conjugatæ <lb/>
sunt eadem: <lb/>
et una est <lb/>
impossibilis: <lb/>
sunt igiture <lb/>
tres tantum <lb/>
diversæ <lb/>
æquationes. <lb/>
et <lb/>
illæ, tres <lb/>
primæ.
<lb/>[<emph style="it">tr:
The conjugate equations are the same, and one is impossible;
there are therefore as many as three different equations, and of these, the first three.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1814" xml:space="preserve">
Ergo sunt tres analogiæ diversæ, et casus 6.
<lb/>[<emph style="it">tr:
Therefore there are three different ratios, and 6 cases.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f491v" o="491v" n="983"/>
<pb file="add_6782_f492" o="492" n="984"/>
<pb file="add_6782_f492v" o="492v" n="985"/>
<pb file="add_6782_f493" o="493" n="986"/>
<pb file="add_6782_f493v" o="493v" n="987"/>
<pb file="add_6782_f494" o="494" n="988"/>
<pb file="add_6782_f494v" o="494v" n="989"/>
<pb file="add_6782_f495" o="495" n="990"/>
<pb file="add_6782_f495v" o="495v" n="991"/>
<pb file="add_6782_f496" o="496" n="992"/>
<pb file="add_6782_f496v" o="496v" n="993"/>
<pb file="add_6782_f497" o="497" n="994"/>
<pb file="add_6782_f497v" o="497v" n="995"/>
<pb file="add_6782_f498" o="498" n="996"/>
<pb file="add_6782_f498v" o="498v" n="997"/>
<pb file="add_6782_f499" o="499" n="998"/>
<pb file="add_6782_f499v" o="499v" n="999"/>
<pb file="add_6782_f500" o="500" n="1000"/>
<pb file="add_6782_f500v" o="500v" n="1001"/>
<pb file="add_6782_f501" o="501" n="1002"/>
<div xml:id="echoid-div327" type="page_commentary" level="2" n="327">
<p>
<s xml:id="echoid-s1815" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1815" xml:space="preserve">
The reference at the top of this page is to Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book IV, Zetetic 4.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum IV <lb/>
Invenire duo triangula rectangula simili datas habentes hypotenusas,
&amp; diducti ab iis tertii trianguli basis, composita ex perpendiculo primi &amp; base secundi,
erit ea quæ præfinitur. <lb/>
Oportebit autem basim illam præfinitam præstare hypotenusæ primi.
</quote>
<lb/>
<quote>
To find two similar right-angled triangles having given hypotenuses,
and subtracting the base of a third triangle,
composed of the perpendicular of the first and the base of the second,
will give a predefined quantity.
The predefined base, moreover, must be greater than the hypotenuse of the first triangle.
</quote>
<lb/>
<s xml:id="echoid-s1816" xml:space="preserve">
Viète used the letters <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>D</mi></mstyle></math> for the hypotenuses of the first and second triangles,
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi></mstyle></math> for the perpendicular of the third.
Harriot followed Viète's working but in his own lower-case notation.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head238" xml:space="preserve" xml:lang="lat">
Zet. 4. 4.
<lb/>[<emph style="it">tr:
Zetetica, Book IV, Zetetic 4.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1818" xml:space="preserve">
operatio fit <lb/>
per. 3am
<lb/>[<emph style="it">tr:
the operation may be done by the third
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f501v" o="501v" n="1003"/>
<pb file="add_6782_f502" o="502" n="1004"/>
<div xml:id="echoid-div328" type="page_commentary" level="2" n="328">
<p>
<s xml:id="echoid-s1819" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1819" xml:space="preserve">
The reference at the top of this page is to Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book IV, Zetetic 2.
This is also Proposition II.9 from the <emph style="it">Arithmetica</emph> of Diophantus,
but Harriot refers only to Viète's version of it.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum II <lb/>
Invenire numero duo quadrata, aequalia duobus aliis datis quadratis.
</quote>
<lb/>
<quote>
To find in numbers two squares equal to two other given squares.
</quote>
<lb/>
<s xml:id="echoid-s1820" xml:space="preserve">
One may interpret the problem as asking for two sets of Pythagorean triples with the same third side,
or two rational right-angled triangles with the same hypotenuse.
Viète called the sides of the first two squares <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>D</mi></mstyle></math>, and the hypotenuse <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>.
The words 'synæreseos' and 'diæreseos' were used by Viète and are typical of the Greek terms
he frequently introduced into his writing.
Harriot followed Viète's working but in his own lower-case notation.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head239" xml:space="preserve" xml:lang="lat">
Zet. lib. 4. 2.
<lb/>[<emph style="it">tr:
Zetetica, Book IV, Zetetic 2.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1822" xml:space="preserve">
prop. 2.
<lb/>[<emph style="it">tr:
Proposition 2
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1823" xml:space="preserve">
via synæreseos
<lb/>[<emph style="it">tr:
by expansion
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1824" xml:space="preserve">
via diæreseos
<lb/>[<emph style="it">tr:
by contraction
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1825" xml:space="preserve">
prop. 1.
<lb/>[<emph style="it">tr:
Proposition 1
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1826" xml:space="preserve">
prop. 3. eadem ac secunda aliter
<lb/>[<emph style="it">tr:
Proposition 3. the same and the second another way.
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f502v" o="502v" n="1005"/>
<pb file="add_6782_f503" o="503" n="1006"/>
<div xml:id="echoid-div329" type="page_commentary" level="2" n="329">
<p>
<s xml:id="echoid-s1827" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1827" xml:space="preserve">
The reference at the top of this page is to Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book IV, Proposition 3.
This is also Proposition II.9 from the <emph style="it">Arithemtica</emph> of Diophantus
(see also Add MS 6782, f. 502).
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum II <lb/>
Rursus, invenire numero duo quadrata, aequalia duobus aliis datis quadratis.
</quote>
<lb/>
<quote>
Again, to find in numbers two squares equal to two other given squares.
</quote>
<lb/>
<s xml:id="echoid-s1828" xml:space="preserve">
One may interpret the problem as asking for two sets of Pythagorean triples with the same third side,
or two rational right-angled triangles with the same hypotenuse.
Viète called the sides of the first two squares <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>D</mi></mstyle></math>.
Harriot followed Viète's working but in his own lower-case notation.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head240" xml:space="preserve" xml:lang="lat">
Zet. lib. 4. 3.
<lb/>[<emph style="it">tr:
Zetetica, Book IV, Zetetic 3.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1830" xml:space="preserve">
1. triang.
<lb/>[<emph style="it">tr:
triangle 1
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1831" xml:space="preserve">
2. Δ. simili.
<lb/>[<emph style="it">tr:
triangle 2, similar
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1832" xml:space="preserve">
3. Δ.
<lb/>[<emph style="it">tr:
triangle 3
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1833" xml:space="preserve">
3. Δ. alterum.
<lb/>[<emph style="it">tr:
triangle 3, alternative
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1834" xml:space="preserve">
3. Δ. aliter.
<lb/>[<emph style="it">tr:
triangle 3 another way
</emph>]<lb/>
</s>
</p>
<pb file="add_6782_f503v" o="503v" n="1007"/>
<pb file="add_6782_f504" o="504" n="1008"/>
<pb file="add_6782_f504v" o="504v" n="1009"/>
<pb file="add_6782_f505" o="505" n="1010"/>
<p xml:lang="lat">
<s xml:id="echoid-s1835" xml:space="preserve">
Talia problemata hic
<emph style="super">schemata explicantur</emph>
<emph style="st">apponuntur</emph> quæ
<emph style="st">conducunt</emph> ad Magisteriorum <lb/>
formas conducunt intelligendas. <emph style="st">[???]</emph> similis.
</s>
</p>
<pb file="add_6782_f505v" o="505v" n="1011"/>
</div>
</text>
</echo>
author	Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date	Wed, 29 Jan 2014 15:57:38 +0100
parents	22d6a63640c6
children	1c8105d0894f