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DE Specs Version 2.1.1 Autumn 2011
author Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date Thu, 02 May 2013 11:29:00 +0200
parents 22d6a63640c6
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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:VWXURW4V.xml</dcterms:identifier>
<dcterms:creator>Harriot, Thomas</dcterms:creator>
<dcterms:title xml:lang="en">Mss. 6783</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/VWXURW4V</echodir>
<log>Automatically generated by bare_xml.py on Tue Nov 15 14:20:53 2011</log>
</metadata>

<text xml:lang="eng" type="free">
<div xml:id="echoid-div1" type="section" level="1" n="1">
<pb file="add_6783_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">
This is the first page of a 11-page treatise on the arithmetic of radicals.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head1" xml:space="preserve" xml:lang="lat">
b.1) De radicalibus
<lb/>[<emph style="it">tr: 
On radicals
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s3" xml:space="preserve">
Invenire plana similia
<lb/>[<emph style="it">tr: 
To find similar planes
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s4" xml:space="preserve">
Invenire similia solida
<lb/>[<emph style="it">tr: 
To find similar solids
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s5" xml:space="preserve">
Invenire similia planoplanum
<lb/>[<emph style="it">tr: 
To find similar plano-planes
</emph>]<lb/>
</s>
</p>
<head xml:id="echoid-head2" xml:space="preserve" xml:lang="">
Additionis exempla
<lb/>[<emph style="it">tr: 
Examples of addition
</emph>]<lb/>
</head>
<pb file="add_6783_f001v" o="1v" n="2"/>
<div xml:id="echoid-div2" type="page_commentary" level="2" n="2">
<p>
<s xml:id="echoid-s6" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s6" xml:space="preserve">
The examples of addition are continued from those on the first page (Add MS 6783, f. 1). <lb/>
The examples are followed by some notes on plane, solid, and plano-plane numbers;
Harriot's notation from the earlier working has been inserted into the translation at the relevant places.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s8" xml:space="preserve">
Additio
<lb/>[<emph style="it">tr: 
Addition
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s9" xml:space="preserve">
Si duo numeri sint similes plani; factus ex illis est quadratus, <lb/>
cuius radix est medium proportionalis, inter datos.
<lb/>[<emph style="it">tr: 
If two numbers [bcdd, bcff] are similar planes, their product is a square
whose root [bcdf] is the mean proportional between the given numbers.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s10" xml:space="preserve">
Si similes plani dividantur per maximum communem divisor, <lb/>
quoti sunt quadrati.
<lb/>[<emph style="it">tr: 
If similar plane numbers [bcdd, bcff]
are divided by their greatest common divisor, the quotients are squares.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s11" xml:space="preserve">
Si duo numeri sint similes solidi; factus e quadrato unius per alterum, <lb/>
est cubus; cuius radix est una medium proportionalis,
<emph style="st">videlicet</emph> <emph style="super">inter datos</emph> <lb/>
et proxima ad illum numerum qui factus fuit quadratus.
<lb/>[<emph style="it">tr: 
If two numbers [bcdfff, bcdggg] are similar solids,
the product of the square of one with the other is a cube,
whose root [bcdffg] is one mean proportional beteen the two given numbers,
and nearest to that number that was made a square.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s12" xml:space="preserve">
Si similes solidi dividantur per maximum communem divisor, <lb/>
quoti sunt cubi.
<lb/>[<emph style="it">tr: 
If similar solid numbers [bcdfff, bcdggg]
are divided by their greatest common divisor, the quotients are cubes.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s13" xml:space="preserve">
Si duo numeri sint similes planoplani; factus e cubo primi <lb/>
per secundum est primum quadrato-quadratum cuius radix est primum <lb/>
medium proportionalis, inter datos.
<lb/>[<emph style="it">tr: 
If two numbers are similar plano-planes [bcdfgggg, bcdfhhhh],
the product of the cube of the first with the second
is a square-square, whose side [bcdfgggh]
is the first mean proportional between the given numbers.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s14" xml:space="preserve">
Si similes plano-plani dividantur per maximum communem divisor, <lb/>
quoti sunt quadrato-quadrati.
<lb/>[<emph style="it">tr: 
If similar plano-plane numbers [bcdfgggg, bcdfhhhh]
are divided by their greatest commmon divisor, the quotients are square-squares.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f002" o="2" n="3"/>
<head xml:id="echoid-head3" xml:space="preserve" xml:lang="lat">
b.2) De radicalibus
<lb/>[<emph style="it">tr: 
On radicals
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s15" xml:space="preserve">
Subductionis exempla
<lb/>[<emph style="it">tr: 
Examples of subtraction
</emph>]<lb/>
<sc>
Here the identity is demonstrated by dividing each term by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>c</mi></mrow></msqrt></mstyle></math>.
</sc>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s16" xml:space="preserve">
Additonis exempla aliter. et similiter <lb/>
subductionis.
<lb/>[<emph style="it">tr: 
Examples of addition another way; and similarly for subtraction.
</emph>]<lb/>
<sc>
Here the identity is demonstrated by dividing each term by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>c</mi><mi>d</mi><mi>d</mi></mrow></msqrt></mstyle></math>.
</sc>
</s>
<lb/>
<s xml:id="echoid-s17" xml:space="preserve">
Sed melius <lb/>
antea.
<lb/>[<emph style="it">tr: 
But it was better before.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f002v" o="2v" n="4"/>
<div xml:id="echoid-div3" type="page_commentary" level="2" n="3">
<p>
<s xml:id="echoid-s18" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s18" xml:space="preserve">
Some numerical examples of addition of radicals.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head4" xml:space="preserve" xml:lang="lat">
Nota.
<lb/>[<emph style="it">tr: 
Note
</emph>]<lb/>
</head>
<pb file="add_6783_f003" o="3" n="5"/>
<div xml:id="echoid-div4" type="page_commentary" level="2" n="4">
<p>
<s xml:id="echoid-s20" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s20" xml:space="preserve">
Further examples of addition and subtraction in numbers and letters (see also Add MS 6783, f. 1 and f.3)
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head5" xml:space="preserve" xml:lang="lat">
b.3. De Additione et Subtractione <emph style="st">(Optime.)</emph>
<lb/>[<emph style="it">tr: 
On addition and subtraction <emph style="st">best</emph>
</emph>]<lb/>
</head>
<pb file="add_6783_f003v" o="3v" n="6"/>
<pb file="add_6783_f004" o="4" n="7"/>
<div xml:id="echoid-div5" type="page_commentary" level="2" n="5">
<p>
<s xml:id="echoid-s22" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s22" xml:space="preserve">
Numerical examples of multiplication of square roots and fourth roots. <lb/>
The final calculation on the page, for example,
is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><msqrt><mrow><mn>4</mn><mn>8</mn></mrow></msqrt></mrow></msqrt><mo>×</mo><msqrt><mrow><msqrt><mrow><mn>3</mn></mrow></msqrt></mrow></msqrt><mo>=</mo><msqrt><mrow><msqrt><mrow><mn>1</mn><mn>4</mn><mn>4</mn></mrow></msqrt></mrow></msqrt><mo>=</mo><msqrt><mrow><mn>1</mn><mn>2</mn></mrow></msqrt></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head6" xml:space="preserve" xml:lang="lat">
b.4. Multiplicatio.
<lb/>[<emph style="it">tr: 
Multiplication
</emph>]<lb/>
</head>
<pb file="add_6783_f004v" o="4v" n="8"/>
<pb file="add_6783_f005" o="5" n="9"/>
<div xml:id="echoid-div6" type="page_commentary" level="2" n="6">
<p>
<s xml:id="echoid-s24" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s24" xml:space="preserve">
Numerical examples of division of square roots and fourth roots. <lb/>
The final calculation on the page, for example, is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>\\frac</mo><mrow><msqrt><mrow><msqrt><mrow><mn>3</mn></mrow></msqrt></mrow></msqrt><mrow><msqrt><mrow><msqrt><mrow><mn>4</mn><mn>8</mn></mrow></msqrt></mrow></msqrt><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head7" xml:space="preserve" xml:lang="lat">
b.5.) Applicatio.
<lb/>[<emph style="it">tr: 
Division
</emph>]<lb/>
</head>
<pb file="add_6783_f005v" o="5v" n="10"/>
<pb file="add_6783_f006" o="6" n="11"/>
<div xml:id="echoid-div7" type="page_commentary" level="2" n="7">
<p>
<s xml:id="echoid-s26" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s26" xml:space="preserve">
Examples of multiplication and division of sums and differences of square roots. <lb/>
The final calculation on the page, for example,
is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>2</mn></mrow><mrow><msqrt><mrow><msqrt><mrow><mn>7</mn></mrow></msqrt><mo>+</mo><msqrt><mrow><mn>3</mn></mrow></msqrt></mrow></msqrt></mrow></mfrac><mo>=</mo><msqrt><mrow><msqrt><mrow><mn>7</mn></mrow></msqrt><mo>-</mo><msqrt><mrow><mn>3</mn></mrow></msqrt></mrow></msqrt></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head8" xml:space="preserve" xml:lang="lat">
b.6.) <lb/>
Multiplicatio.
<lb/>[<emph style="it">tr: 
Multiplication
</emph>]<lb/>
</head>
<head xml:id="echoid-head9" xml:space="preserve" xml:lang="lat">
Applicatio.
<lb/>[<emph style="it">tr: 
Division
</emph>]<lb/>
</head>
<pb file="add_6783_f006v" o="6v" n="12"/>
<pb file="add_6783_f007" o="7" n="13"/>
<div xml:id="echoid-div8" type="page_commentary" level="2" n="8">
<p>
<s xml:id="echoid-s28" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s28" xml:space="preserve">
Further examples, continuing from f. 6, of division of sums and differences of square roots. <lb/>
In particular, the calculation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><msqrt><mrow><mn>8</mn></mrow></msqrt></mrow><mrow><msqrt><mrow><mn>3</mn></mrow></msqrt><mo>+</mo><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></math> is worked in two ways.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head10" xml:space="preserve" xml:lang="lat">
b.7.) Applicatio.
<lb/>[<emph style="it">tr: 
Division
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s30" xml:space="preserve">
Aliter.
<lb/>[<emph style="it">tr: 
Another way
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f007v" o="7v" n="14"/>
<pb file="add_6783_f008" o="8" n="15"/>
<div xml:id="echoid-div9" type="page_commentary" level="2" n="9">
<p>
<s xml:id="echoid-s31" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s31" xml:space="preserve">
Calculation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>f</mi><mi>g</mi><mo>+</mo><mi>b</mi><mi>d</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>f</mi><mi>g</mi></mrow></msqrt></mstyle></math>. <lb/>
See also Add MS 6783, f. 9, where similar working is repeated, but is rather easier to follow.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head11" xml:space="preserve" xml:lang="lat">
b.8. Radicum extractio.
<lb/>[<emph style="it">tr: 
Extraction of roots
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s33" xml:space="preserve">
1. 2. 3. significant <lb/>
ordinem proportionalium <lb/>
in diagramatis.
<lb/>[<emph style="it">tr: 
1, 2, 3, signify the order of the proportions in the diagram.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s34" xml:space="preserve">
Ergo radix.
<lb/>[<emph style="it">tr: 
Therefore the root.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f008v" o="8v" n="16"/>
<pb file="add_6783_f009" o="9" n="17"/>
<div xml:id="echoid-div10" type="page_commentary" level="2" n="10">
<p>
<s xml:id="echoid-s35" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s35" xml:space="preserve">
Calculation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>f</mi><mi>g</mi><mo>+</mo><mi>b</mi><mi>d</mi></mrow></msqrt></mstyle></math>. <lb/>
This is an algebraic version of the problem posed in Euclid X.59, to find the root of a sixth binome. <lb/>
The same problem was also worked on Add MS 6783, f. 7,
but <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>f</mi><mi>g</mi><mo>+</mo><mi>b</mi><mi>d</mi><mo maxsize="1">)</mo></mstyle></math> there has been replaced by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math> here. <lb/>
Harriot's method, following Euclid, is to construct the square shown in the second diagram,
in which the area of squares 1 + 3 represents <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>c</mi><mi>d</mi></mrow></msqrt></mstyle></math>
and the area of rectangles 2 + 2 represent <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>g</mi></mstyle></math> (so each is of area <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mfrac><mrow><mi>f</mi><mi>g</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msqrt></mstyle></math>.
The side of square 1 is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/>
It is clear from simple geometry that area 1 : area 2 = area 2 : area 3,
so Harriot can write down three proportional terms
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>,</mo><msqrt><mrow><mfrac><mrow><mi>f</mi><mi>g</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msqrt><mo>,</mo><msqrt><mrow><mi>c</mi><mi>d</mi></mrow></msqrt><mo>-</mo><mi>a</mi></mstyle></math>.
Thus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi><mi>g</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo>=</mo><msqrt><mrow><mi>c</mi><mi>d</mi><mi>a</mi><mi>a</mi></mrow></msqrt><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>, leading to a quadratic equation in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/>
Harriot solves this equation by completing the square, giving him two possible values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>.
The side of the whole square is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>a</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><msqrt><mrow><mi>c</mi><mi>d</mi></mrow></msqrt><mo>-</mo><mi>a</mi></mrow></msqrt></mstyle></math>.
Using either of his values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, he obtains the expression seen next to the words  18Ergo Radix'.
Squaring this expression gives him, as required, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>c</mi><mi>d</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>f</mi><mi>g</mi></mrow></msqrt></mstyle></math>, the area of the whole square.
He also observes that the sum of the roots gives him <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>c</mi><mi>d</mi></mrow></msqrt></mstyle></math>. <lb/>
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head12" xml:space="preserve" xml:lang="lat">
b.9.) Radicum extractio.
<lb/>[<emph style="it">tr: 
Extraction of roots
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s37" xml:space="preserve">
Ergo radix.
<lb/>[<emph style="it">tr: 
Therefore the root
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f009v" o="9v" n="18"/>
<pb file="add_6783_f010" o="10" n="19"/>
<div xml:id="echoid-div11" type="page_commentary" level="2" n="11">
<p>
<s xml:id="echoid-s38" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s38" xml:space="preserve">
Calculation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>d</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>b</mi><mi>c</mi></mrow></msqrt></mstyle></math>. <lb/>
See Add MS 6783, f. 9 for the working of the same problem in slightly different notation.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head13" xml:space="preserve" xml:lang="lat">
b.10.) Radicum extractio.
<lb/>[<emph style="it">tr: 
Extraction of roots
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s40" xml:space="preserve">
1. 2. 3. <lb/>
Hæ numeri non <lb/>
significant quantitates <lb/>
sed ordinem propor-<lb/>
tionalium.
<lb/>[<emph style="it">tr: 
1, 2, 3. These numbers do not signify quantities but the order of the proportions.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s41" xml:space="preserve">
Radix.
<lb/>[<emph style="it">tr: 
Root
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f010v" o="10v" n="20"/>
<pb file="add_6783_f011" o="11" n="21"/>
<div xml:id="echoid-div12" type="page_commentary" level="2" n="12">
<p>
<s xml:id="echoid-s42" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s42" xml:space="preserve">
A binome is a quantity 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>. <lb/>
On this page Harriot applies the method shown on Add MS 6783, f. 9 to find the square root of
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mfrac><mrow><mi>b</mi><mi>c</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>d</mi><mi>c</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msqrt><mo>+</mo><msqrt><mrow><mfrac><mrow><mi>b</mi><mi>c</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>d</mi><mi>c</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msqrt></mstyle></math>.
The term 'universal' here signifies that he is taking the root of the entire quantity.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head14" xml:space="preserve" xml:lang="lat">
b.11.) Radix binomij universalis.
<lb/>[<emph style="it">tr: 
The universal root of a binome
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s44" xml:space="preserve">
non <lb/>
Binomium <lb/>
proprie
<lb/>[<emph style="it">tr: 
not a proper binome
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f011v" o="11v" n="22"/>
<div xml:id="echoid-div13" type="page_commentary" level="2" n="13">
<p>
<s xml:id="echoid-s45" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s45" xml:space="preserve">
Here Harriot gives a general identity
(which follows from his work on Add MS 6783, f. 9 and elsewhere in this section)
for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>2</mn><mi>b</mi><mo>+</mo><mn>2</mn><mi>c</mi></mrow></msqrt></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head15" xml:space="preserve" xml:lang="lat">
Radicis Binomij, species universalis. <lb/>
canon
<lb/>[<emph style="it">tr: 
For the root of a binome, the general case, a rule.
</emph>]<lb/>
</head>
<pb file="add_6783_f012" o="12" n="23"/>
<pb file="add_6783_f012v" o="12v" n="24"/>
<pb file="add_6783_f013" o="13" n="25"/>
<pb file="add_6783_f013v" o="13v" n="26"/>
<pb file="add_6783_f014" o="14" n="27"/>
<pb file="add_6783_f014v" o="14v" n="28"/>
<pb file="add_6783_f015" o="15" n="29"/>
<pb file="add_6783_f015v" o="15v" n="30"/>
<pb file="add_6783_f016" o="16" n="31"/>
<pb file="add_6783_f016v" o="16v" n="32"/>
<pb file="add_6783_f017" o="17" n="33"/>
<pb file="add_6783_f017v" o="17v" n="34"/>
<pb file="add_6783_f018" o="18" n="35"/>
<pb file="add_6783_f018v" o="18v" n="36"/>
<pb file="add_6783_f019" o="19" n="37"/>
<pb file="add_6783_f019v" o="19v" n="38"/>
<pb file="add_6783_f020" o="20" n="39"/>
<div xml:id="echoid-div14" type="page_commentary" level="2" n="14">
<p>
<s xml:id="echoid-s47" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s47" xml:space="preserve">
This folio refers to page 44v of Michael Stifel, <emph style="it">Arithmetica integra</emph> (1544).
The relevant passage from Stifel is as follows.
</s>
<lb/>
<quote xml:lang="lat">
De inventione numerorum, qui peculiariter pertinent ad suas species extractionum. <lb/>
Restat iam ut tradam modum inueniendi numeros, qui peculiariter pertinent ad quam libet speciem extractionum,
quatenus perfecta habeatur &amp; absoluta huius negotij consumatio. Tradam autem huius modi inuentionem,
per tabulam sequentem, quae ut in infinitum extendatur cuipse facile uidebis,
quam primum uideris rationem qua construitur. Sic autem constructam uides.
<lb/>
<lb/>[<emph style="it">tr: 
On finding numbers which pertain particularly to the extraction of roots. <lb/>
It remains for me to teach how to find numbers, which pertain particularly to extracting any root,
that this matter might be perfected and wholly completed.
Moreover, I teach this method of finding numbers by the following table,
which may be extended indefinitely as you can easily see, once you have seen how it is constructed.
Thus moreover you see it constructed.
</emph>]<lb/>
</quote>
<lb/>
<s xml:id="echoid-s48" xml:space="preserve">
Stifel goes on to explain that the numbers in the second and subsequent columns
are obtained as the sum of two entries from the preceding row. <lb/>
Harriot's table is identical to Stifel's except that he has extended the rows further to the right.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s50" xml:space="preserve">
Stifelius pag. 44. b. <lb/>
Numeri ad extractionem <lb/>
Radicum.
<lb/>[<emph style="it">tr: 
Stifel, page 44v <lb/>
Numbers for the extraction of roots.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f020v" o="20v" n="40"/>
<pb file="add_6783_f021" o="21" n="41"/>
<pb file="add_6783_f021v" o="21v" n="42"/>
<pb file="add_6783_f022" o="22" n="43"/>
<pb file="add_6783_f022v" o="22v" n="44"/>
<pb file="add_6783_f023" o="23" n="45"/>
<pb file="add_6783_f023v" o="23v" n="46"/>
<pb file="add_6783_f024" o="24" n="47"/>
<pb file="add_6783_f024v" o="24v" n="48"/>
<pb file="add_6783_f025" o="25" n="49"/>
<pb file="add_6783_f025v" o="25v" n="50"/>
<pb file="add_6783_f026" o="26" n="51"/>
<pb file="add_6783_f026v" o="26v" n="52"/>
<pb file="add_6783_f027" o="27" n="53"/>
<pb file="add_6783_f027v" o="27v" n="54"/>
<pb file="add_6783_f028" o="28" n="55"/>
<pb file="add_6783_f028v" o="28v" n="56"/>
<pb file="add_6783_f029" o="29" n="57"/>
<div xml:id="echoid-div15" type="page_commentary" level="2" n="15">
<p>
<s xml:id="echoid-s51" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s51" xml:space="preserve">
This page shows the nmbers 1 to 31 written as sums of 1, 2, 4, 8, and 16.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6783_f029v" o="29v" n="58"/>
<pb file="add_6783_f030" o="30" n="59"/>
<pb file="add_6783_f030v" o="30v" n="60"/>
<pb file="add_6783_f031" o="31" n="61"/>
<pb file="add_6783_f031v" o="31v" n="62"/>
<pb file="add_6783_f032" o="32" n="63"/>
<pb file="add_6783_f032v" o="32v" n="64"/>
<pb file="add_6783_f033" o="33" n="65"/>
<pb file="add_6783_f033v" o="33v" n="66"/>
<pb file="add_6783_f034" o="34" n="67"/>
<pb file="add_6783_f034v" o="34v" n="68"/>
<pb file="add_6783_f035" o="35" n="69"/>
<div xml:id="echoid-div16" type="page_commentary" level="2" n="16">
<p>
<s xml:id="echoid-s53" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s53" xml:space="preserve">
This page appears to be an exploration of recurring decimals. Some of the calculations shown are
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn></mrow><mrow><mn>7</mn></mrow></mfrac><mo>=</mo><mn>1</mn><mn>4</mn><mn>2</mn><mn>8</mn><mn>5</mn><mn>7</mn><mn>1</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn><mn>0</mn><mn>0</mn></mrow><mrow><mn>4</mn><mn>9</mn></mrow></mfrac><mo>=</mo><mn>2</mn><mo>.</mo><mn>0</mn><mn>4</mn><mn>0</mn><mn>8</mn><mn>1</mn><mn>6</mn><mn>3</mn><mn>2</mn><mo>.</mo><mo>.</mo><mo>.</mo></mstyle></math>,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mfrac><mrow><mn>1</mn><mn>0</mn></mrow><mrow><mn>7</mn></mrow></mfrac></mrow></msqrt><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn><mn>1</mn><mn>9</mn><mn>5</mn><mn>2</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mfrac><mrow><mn>1</mn><mn>0</mn><mn>0</mn></mrow><mrow><mn>7</mn></mrow></mfrac></mrow></msqrt><mo>=</mo><mn>3</mn><mo>.</mo><mn>7</mn><mn>7</mn><mn>9</mn><mn>6</mn><mo>.</mo><mo>.</mo><mo>.</mo></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head16" xml:space="preserve" xml:lang="lat">
Nota subtilitates.
<lb/>[<emph style="it">tr: 
An examination of exactness.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s55" xml:space="preserve">
A. post priorem valorem <lb/>
figuræ. vel.
<lb/>[<emph style="it">tr: 
A. after the first value of the figure, or,
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f035v" o="35v" n="70"/>
<pb file="add_6783_f036" o="36" n="71"/>
<p>
<s xml:id="echoid-s56" xml:space="preserve">
1)
</s>
</p>
<head xml:id="echoid-head17" xml:space="preserve" xml:lang="lat">
De reductione æquationum
<lb/>[<emph style="it">tr: 
On the reduction of equations
</emph>]<lb/>
</head>
<pb file="add_6783_f036v" o="36v" n="72"/>
<pb file="add_6783_f037" o="37" n="73"/>
<div xml:id="echoid-div17" type="page_commentary" level="2" n="17">
<p>
<s xml:id="echoid-s57" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s57" xml:space="preserve">This sheet refers to pages 411 and 359 of Stevin's
<emph style="it">L'arithmétique ... aussi l'algebre</emph> (1585).
Similar questions appear on both pages.
</s>
<lb/>
<s xml:id="echoid-s58" xml:space="preserve">
page 339
</s>
<lb/>
<quote xml:lang="fra">
Il nous faut doncques trouuer deux nombres tells, que le quarré de la moitie du premier soit egal au second,
&amp; que la produict du premier par le second + 3 soit 225.
<lb/>[<emph style="it">tr: 
We must therefore find two numbers such that the square of half the first is equal to the second,
and the product of the first and the second, plus 3, is 225.
</emph>]<lb/>
</quote>
<lb/>
<s xml:id="echoid-s59" xml:space="preserve">
page 411
</s>
<lb/><quote xml:lang="fra">
Question XII. <lb/>
Trouuons deux nombres tells, que le quarré de la moitie du premier soit egal au second,
&amp; que le produict du premier par le second + 16, soit 225.
<lb/>[<emph style="it">tr: 
Let us fnd two numbers, such that the square of half the first is eual to the second,
amd the product of the first and the second, plus 16, is 225.
</emph>]<lb/>
</quote>
<lb/>
<s xml:id="echoid-s60" xml:space="preserve">
Stevin's questions are posed with specific numbers, but Harriot examines the question in gneral. <lb/>
For the sheet paginated 1) that precedes this one, see Add MS 6783, f. 36.
See also Add MS 6783, f. 38, f. 39, f.39v, f. 42, and also Add MS 6783, f. 89 for another problem of the same kind.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s62" xml:space="preserve">
2) Stevin. pag. 411. et 359.
</s>
</p>
<pb file="add_6783_f037v" o="37v" n="74"/>
<pb file="add_6783_f038" o="38" n="75"/>
<pb file="add_6783_f038v" o="38v" n="76"/>
<pb file="add_6783_f039" o="39" n="77"/>
<pb file="add_6783_f039v" o="39v" n="78"/>
<pb file="add_6783_f040" o="40" n="79"/>
<div xml:id="echoid-div18" type="page_commentary" level="2" n="18">
<p>
<s xml:id="echoid-s63" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s63" xml:space="preserve">
This sheet refers to Stevin's <emph style="it">L'arithmétique ... aussi l'algebre</emph> (1585).
Pages 269 to 280 contain a section entitled 'De la reduction' on reducing, or simplifying, equations.
On page 271, Stevin gives the first rule of reduction:
</s>
<lb/>
<quote xml:lang="fra">
Reigle I. <lb/>
Si les deux termes egaux donnez n'eussent pas l'inferieure quantité (0), on posera qu'il le soit,
delaissant son signe de plus haulte quantité, &amp; chascune des autres quantitez, descendra par egale distance. <lb/>
<lb/>[<emph style="it">tr: 
Rule I. <lb/>
If the two given equal sides do not contain the lowest quantity (of degree 0), one makes it so,
lowering the degree of the highest quantity, and each of the other quantities will be lowered by the same amount.
</emph>]<lb/>
</quote>
<lb/>
<s xml:id="echoid-s64" xml:space="preserve">
On page 272, Stevin gives his second and third rules of reduction:
</s>
<lb/>
<quote xml:lang="fra">
Reigle II. <lb/>
Pour reduire le nombre de multitude de la superieure quantité en vnité,
on divisera par lui toutes les quantitez proposées. <lb/>
Reigle III. <lb/>
Si les deux terms egaux eussent quantitez de mesme hauteur,
on les ostera toutes deux, ou l'vne, par le moien d'egale addition, ou d'egale soubstraction.
<lb/>[<emph style="it">tr: 
Rule II. <lb/>
To reduce the number of the highest quantitie to one,
one divides by that all the proposed quantities. <lb/>
Rule III. <lb/>
If the two equal sides have quantities of the same degree,
one can get rid of both, or one of them, by means of equal addition or equal subtraction.
</emph>]<lb/>
</quote>
<lb/>
<s xml:id="echoid-s65" xml:space="preserve">
For each rule Stevin offered examples. Harriot has re-worked all of them in his own notation.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s67" xml:space="preserve">
Stevin. pa.271.
</s>
</p>
<pb file="add_6783_f040v" o="40v" n="80"/>
<pb file="add_6783_f041" o="41" n="81"/>
<pb file="add_6783_f041v" o="41v" n="82"/>
<pb file="add_6783_f042" o="42" n="83"/>
<div xml:id="echoid-div19" type="page_commentary" level="2" n="19">
<p>
<s xml:id="echoid-s68" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s68" xml:space="preserve">
This sheet refers to the problem on page 411 of Stevin's
<emph style="it">L'arithmétique ... aussi l'algebre</emph> (1585).
</s>
<lb/>
<quote xml:lang="fra">
Question XII. <lb/>
Trouuons deux nombres tells, que le quarré de la moitie du premier soit egal au second,
&amp; que le produict du premier par le second + 16, soit 225.
<lb/>[<emph style="it">tr: 
Question XII. <lb/>
Find two numbers, such that the quare of half the first is equal to the second,
and the product of the first and the second, plus 16, is 225.
</emph>]<lb/>
</quote>
<lb/>
<s xml:id="echoid-s69" xml:space="preserve">
Harriot worked this problem in general form in Add MS 6783, f. 37 and f. 89.
Here he investigates it using the technique of multiplying the root.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s71" xml:space="preserve">
Stevin. pag. 411.
</s>
</p>
<p>
<s xml:id="echoid-s72" xml:space="preserve">
De multiplicatione radicum
<lb/>[<emph style="it">tr: 
On the multiplication of roots
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f042v" o="42v" n="84"/>
<pb file="add_6783_f043" o="43" n="85"/>
<pb file="add_6783_f043v" o="43v" n="86"/>
<pb file="add_6783_f044" o="44" n="87"/>
<pb file="add_6783_f044v" o="44v" n="88"/>
<div xml:id="echoid-div20" type="page_commentary" level="2" n="20">
<p>
<s xml:id="echoid-s73" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s73" xml:space="preserve">
The table at the top right shows the numbers of Pascal's triangle set out in their simplest form. <lb/> <lb/>
The table at the top left shows the same numbers written as fractions,
thus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>4</mn><mn>5</mn><mn>6</mn></mrow><mrow><mn>1</mn><mn>2</mn><mn>3</mn></mrow></mfrac></mstyle></math> on the diagonal is to be read as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>4</mn><mo>×</mo><mn>5</mn><mo>×</mo><mn>6</mn></mrow><mrow><mn>1</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>3</mn></mrow></mfrac><mo>=</mo><mn>2</mn><mn>0</mn></mstyle></math>,
and similarly for all the rest. <lb/> <lb/>
The third table displays the same numbers in a different arrangement,
and suggests how they arise in the study of combinations. <lb/>
Thus the first margin note is <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,
the number of ways of choosing <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> objects from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>.
This is always 1, which appears repeatedly along the lowest diagonal. <lb/>
The second margin note is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></mfrac></mstyle></math>,
the number of ways of choosing 1 object from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>.
The fractions <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></mfrac></mstyle></math> are seen along the second diagonal. <lb/>
The third margin note is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi><mo maxsize="1">(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo maxsize="1">)</mo></mrow><mrow><mn>1</mn><mo>×</mo><mn>2</mn></mrow></mfrac></mstyle></math>,
the number of ways of choosing 2 objects from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>.
The fractions <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> are seen along the third diagonal. <lb/>
The fourth margin note is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi><mo maxsize="1">(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo maxsize="1">)</mo><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>,
the number of ways of choosing 3 objects from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>.
The fractions <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> are seen along the fourth diagonal. <lb/>
These rules were known to Harriot from his study of Girolamo Cardano,
<emph style="it">Opus novum de proportionibus</emph> (1570), propositions 137 and 170
(see Add MS 6782, f. 44). <lb/> <lb/>
The fourth table displays the numbers in the same layout as the third table, but now explicitly calculated.
The numbers 1, 2, 4, 8, 16, ... below it are the sums for each column.
A note on the right observes that the sum for each column is twice that of the preceding one. <lb/>
This sheet continues with third and fourth notations for triangular numbers on Add MS 6782, f. 330.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head18" 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-s75" xml:space="preserve">
1<emph style="super">a</emph>. Notatio triangularium <lb/>
in notis singularibus <lb/>
seu assumptis sive <lb/>
usitatis
<lb/>[<emph style="it">tr: 
1st. Notation for the triangular numbers in figures alone, as first taken up, or as usual.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s76" xml:space="preserve">
2<emph style="super">a</emph>. notatio [triangular]ium <lb/>
in notis specialibus
<lb/>[<emph style="it">tr: 
2nd. Notation for the triangular numbers in a particular way.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s77" xml:space="preserve">
3<emph style="super">a</emph> notatio <lb/>
generalis in <lb/>
alia charta. <lb/>
et 4<emph style="super">a</emph>.
<lb/>[<emph style="it">tr: 
A third and general notation is in another sheet, and a fourth.
</emph>]<lb/>
<sc>
The third and fourth notations are to be found on Add MS 6782, f. 330.
</sc>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s78" xml:space="preserve">
Ad combinationes
<lb/>[<emph style="it">tr: 
For combinations
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s79" xml:space="preserve">
Ergo: <lb/>
Qualibet series perpendicularis <lb/>
dupla est precedentium.
<lb/>[<emph style="it">tr: 
Therefore, any perpendicular series is twice the preceding one.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f045" o="45" n="89"/>
<div xml:id="echoid-div21" type="page_commentary" level="2" n="21">
<p>
<s xml:id="echoid-s80" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s80" xml:space="preserve">
Some linear equations, context unidentified.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6783_f045v" o="45v" n="90"/>
<pb file="add_6783_f046" o="46" n="91"/>
<pb file="add_6783_f046v" o="46v" n="92"/>
<pb file="add_6783_f047" o="47" n="93"/>
<pb file="add_6783_f047v" o="47v" n="94"/>
<pb file="add_6783_f048" o="48" n="95"/>
<pb file="add_6783_f048v" o="48v" n="96"/>
<pb file="add_6783_f049" o="49" n="97"/>
<div xml:id="echoid-div22" type="page_commentary" level="2" n="22">
<p>
<s xml:id="echoid-s82" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s82" xml:space="preserve">
A list of quartic equations and their roots, some of them imaginary.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head19" xml:space="preserve" xml:lang="lat">
Æquationum Exempla
<lb/>[<emph style="it">tr: 
Examples of equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s84" xml:space="preserve">
recte
<lb/>[<emph style="it">tr: 
correct
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f049v" o="49v" n="98"/>
<pb file="add_6783_f050" o="50" n="99"/>
<div xml:id="echoid-div23" type="page_commentary" level="2" n="23">
<p>
<s xml:id="echoid-s85" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s85" xml:space="preserve">
Some quartic equations and their roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s87" xml:space="preserve">
recte
<lb/>[<emph style="it">tr: 
correct
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f050v" o="50v" n="100"/>
<div xml:id="echoid-div24" type="page_commentary" level="2" n="24">
<p>
<s xml:id="echoid-s88" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s88" xml:space="preserve">
Some quartic equations and their roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s90" xml:space="preserve">
recte
<lb/>[<emph style="it">tr: 
correct
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f051" o="51" n="101"/>
<div xml:id="echoid-div25" type="page_commentary" level="2" n="25">
<p>
<s xml:id="echoid-s91" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s91" xml:space="preserve">
Some quartic equations and their roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s93" xml:space="preserve">
recte
<lb/>[<emph style="it">tr: 
correct
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f051v" o="51v" n="102"/>
<pb file="add_6783_f052" o="52" n="103"/>
<div xml:id="echoid-div26" type="page_commentary" level="2" n="26">
<p>
<s xml:id="echoid-s94" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s94" xml:space="preserve">
Some quartic equations and their roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s96" xml:space="preserve">
recte
<lb/>[<emph style="it">tr: 
correct
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s97" xml:space="preserve">
an plures quære.
<lb/>[<emph style="it">tr: 
investigate whether it has more.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s98" xml:space="preserve">
(habet
<lb/>[<emph style="it">tr: 
it has
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f052v" o="52v" n="104"/>
<div xml:id="echoid-div27" type="page_commentary" level="2" n="27">
<p>
<s xml:id="echoid-s99" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s99" xml:space="preserve">
Some quartic equations and their roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s101" xml:space="preserve">
recte
<lb/>[<emph style="it">tr: 
correct
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f053" o="53" n="105"/>
<div xml:id="echoid-div28" type="page_commentary" level="2" n="28">
<p>
<s xml:id="echoid-s102" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s102" xml:space="preserve">
Some quartic equations and their roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s104" xml:space="preserve">
recte
<lb/>[<emph style="it">tr: 
correct
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s105" xml:space="preserve">
recipro-<lb/>
ca
<lb/>[<emph style="it">tr: 
reciprocal
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f053v" o="53v" n="106"/>
<div xml:id="echoid-div29" type="page_commentary" level="2" n="29">
<p>
<s xml:id="echoid-s106" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s106" xml:space="preserve">
Some quartic equations and their roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s108" xml:space="preserve">
recte
<lb/>[<emph style="it">tr: 
correct
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f054" o="54" n="107"/>
<div xml:id="echoid-div30" type="page_commentary" level="2" n="30">
<p>
<s xml:id="echoid-s109" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s109" xml:space="preserve">
Some quartic equations and their roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s111" xml:space="preserve">
recte
<lb/>[<emph style="it">tr: 
correct
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f054v" o="54v" n="108"/>
<pb file="add_6783_f055" o="55" n="109"/>
<pb file="add_6783_f055v" o="55v" n="110"/>
<pb file="add_6783_f056" o="56" n="111"/>
<div xml:id="echoid-div31" type="page_commentary" level="2" n="31">
<p>
<s xml:id="echoid-s112" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s112" xml:space="preserve">
Some quartic equations and their roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s114" xml:space="preserve">
recte
<lb/>[<emph style="it">tr: 
correct
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s115" xml:space="preserve">
K. ista differentia semper habet duas radices affirmatus <lb/>
ut ex canone trinomia reductione apparet <lb/>
si non fallor, et ideo coniugata semper negativus.
<lb/>[<emph style="it">tr: 
K. this type always has two positive roots, as is clear from the canon for the reduction of trinomials
if I am not mistaken, and therefore the conjugate always has negatives.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f056v" o="56v" n="112"/>
<head xml:id="echoid-head20" xml:space="preserve">
f.8
</head>
<pb file="add_6783_f057" o="57" n="113"/>
<div xml:id="echoid-div32" type="page_commentary" level="2" n="32">
<p>
<s xml:id="echoid-s116" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s116" xml:space="preserve">
Some quartic equations and their roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s118" xml:space="preserve">
recte
<lb/>[<emph style="it">tr: 
correct
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s119" xml:space="preserve">
Harum <emph style="st">Illarum</emph> coniugatæ habent unicam radicum
<emph style="st">hypostatic</emph> <emph style="super">affirmatum</emph> <lb/>
nimirum illam quæ hic negativa est.
<lb/>[<emph style="it">tr: 
The conjugates of these <emph style="st">those</emph> have a single positive root, clearly that which is here negative.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s120" xml:space="preserve">
recte
<lb/>[<emph style="it">tr: 
correct
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f057v" o="57v" n="114"/>
<head xml:id="echoid-head21" xml:space="preserve">
f.8
</head>
<p xml:lang="lat">
<s xml:id="echoid-s121" xml:space="preserve">
sit æquationis forma data: <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mo>+</mo><mi>f</mi><mi>f</mi><mi>g</mi><mi>a</mi><mo>+</mo><mi>c</mi><mi>d</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>4</mn><mi>r</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>. <lb/>
quæ quatuor radicum potis est: oportet invenire <lb/>
æquationem sub eadem froma <emph style="st">quæ</emph> <emph style="super">ut</emph> habeat solummodo
<emph style="super">radices</emph> duas. <lb/>
Quoniam potestas est negativa non feret unicam vel tres [???]
<lb/>[<emph style="it">tr: 
Let the equation be in the given form <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mo>+</mo><mi>f</mi><mi>f</mi><mi>g</mi><mi>a</mi><mo>+</mo><mi>c</mi><mi>d</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>4</mn><mi>r</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>. <lb/>
which may have four roots;
it is required to find an equation of the same form that has only two roots.
Because the highest power is negative, if cannot have one or three.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f058" o="58" n="115"/>
<pb file="add_6783_f058v" o="58v" n="116"/>
<pb file="add_6783_f059" o="59" n="117"/>
<div xml:id="echoid-div33" type="page_commentary" level="2" n="33">
<p>
<s xml:id="echoid-s122" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s122" xml:space="preserve">
Some linear, quadratic, and cubic equations and their roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6783_f059v" o="59v" n="118"/>
<pb file="add_6783_f060" o="60" n="119"/>
<div xml:id="echoid-div34" type="page_commentary" level="2" n="34">
<p>
<s xml:id="echoid-s124" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s124" xml:space="preserve">
Some cubic equations and their roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6783_f060v" o="60v" n="120"/>
<pb file="add_6783_f061" o="61" n="121"/>
<div xml:id="echoid-div35" type="page_commentary" level="2" n="35">
<p>
<s xml:id="echoid-s126" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s126" xml:space="preserve">
See also Add MS 6783, f. 35.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head22" xml:space="preserve" xml:lang="lat">
Nota subtilitates.
<lb/>[<emph style="it">tr: 
An examination of exactness.
</emph>]<lb/>
</head>
<pb file="add_6783_f061v" o="61v" n="122"/>
<div xml:id="echoid-div36" type="page_commentary" level="2" n="36">
<p>
<s xml:id="echoid-s128" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s128" xml:space="preserve">
An analysis of the letters in the words at the beginning of the Book of Genesis:
</s>
<lb/>
<quote>
In the beginning God created the heaven and the earth <lb/>
and the earth was without form and voyde and darkness was upon ...
</quote>
<lb/>
<s xml:id="echoid-s129" xml:space="preserve">
Harriot observes, for example, that the letter 'a' occurs in line 1 in places 21. 30, 34, 41;
and in line 2 in places 1, 8, 13, 27, etc.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6783_f062" o="62" n="123"/>
<div xml:id="echoid-div37" type="page_commentary" level="2" n="37">
<p>
<s xml:id="echoid-s131" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s131" xml:space="preserve">
The continuation of Add MS 6783, f. 62v.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s133" xml:space="preserve">
Nota 5<emph style="super">o</emph>.
<lb/>[<emph style="it">tr: 
Note 5.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s134" xml:space="preserve">
Quoniam <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> ex data æquatione, <lb/>
erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>&gt;</mo><mi>a</mi></mstyle></math> videlicet radice qualibet.
<lb/>[<emph style="it">tr: 
Because <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> in the given equaion, evidently <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>&gt;</mo><mi>a</mi></mstyle></math> for any root.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s135" xml:space="preserve">
ergo si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>=</mo><mn>3</mn><mn>7</mn><mn>0</mn></mstyle></math> <lb/>
erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mn>7</mn><mn>0</mn><mo>&gt;</mo></mstyle></math> qualibet radix.
<lb/>[<emph style="it">tr: 
Therefore if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>=</mo><mn>3</mn><mn>7</mn><mn>0</mn></mstyle></math>, it must be that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mn>7</mn><mn>0</mn><mo>&gt;</mo></mstyle></math> any root.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s136" xml:space="preserve">
ergo qualibet radix in data æquatione numerorum <lb/>
non constabit ex pluribus figuris quam tribus.
<lb/>[<emph style="it">tr: 
Therefore any root of the given equation in numbers cannot consist of more figures than three.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f062v" o="62v" n="124"/>
<div xml:id="echoid-div38" type="page_commentary" level="2" n="38">
<p>
<s xml:id="echoid-s137" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s137" xml:space="preserve">
Another version of Add MS 6782, f. 417.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head23" 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-s139" 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-s140" 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-s141" 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-s142" 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-s143" 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-s144" 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-s145" 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-s146" xml:space="preserve">
Ad limites radicum.
<lb/>[<emph style="it">tr: 
For the limits of the roots.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s147" xml:space="preserve">
Nota 1<emph style="super">o</emph>.
<lb/>[<emph style="it">tr: 
Note 1.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s148" xml:space="preserve">
Si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> sit æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</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></mstyle></math> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s149" 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/>[<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-s150" xml:space="preserve">
Nota 2<emph style="super">o</emph>.
<lb/>[<emph style="it">tr: 
Note 2.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s151" xml:space="preserve">
Nota 3<emph style="super">o</emph>.
<lb/>[<emph style="it">tr: 
Note 3.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s152" xml:space="preserve">
Dico quod:
<lb/>[<emph style="it">tr: 
I say that:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s153" xml:space="preserve">
ponatur:
<lb/>[<emph style="it">tr: 
supposing:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s154" 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-s155" xml:space="preserve">
Nota 4<emph style="super">o</emph>.
<lb/>[<emph style="it">tr: 
Note 4.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s156" xml:space="preserve">
Si una radix sit cognita <lb/>
altera erit nota. <lb/>[<emph style="it">tr: 
If one root is known, the other will be known.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s157" xml:space="preserve">
Nimirum: sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> cognita.
<lb/>[<emph style="it">tr: 
Evidently, if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is known.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s158" xml:space="preserve">
Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> cognita.
<lb/>[<emph style="it">tr: 
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> be known.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s159" 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>8</mn></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s160" xml:space="preserve">
verte
<lb/>[<emph style="it">tr: 
turn over
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f063" o="63" n="125"/>
<pb file="add_6783_f063v" o="63v" n="126"/>
<pb file="add_6783_f064" o="64" n="127"/>
<pb file="add_6783_f064v" o="64v" n="128"/>
<pb file="add_6783_f065" o="65" n="129"/>
<head xml:id="echoid-head24" xml:space="preserve" xml:lang="lat">
e. De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s161" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda.
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s162" xml:space="preserve">
sed hic non fit <lb/>
quia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>&lt;</mo><mi>r</mi></mstyle></math>.
</s>
<lb/>
<s xml:id="echoid-s163" xml:space="preserve">
erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>-</mo><mi>p</mi><mo>=</mo><mi>a</mi></mstyle></math>, <emph style="super">radici</emph> <lb/>
maior adventitiæ <lb/>
propositæ.
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s164" xml:space="preserve">
fiat antithesis, ita:
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s165" xml:space="preserve">
Tum: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>-</mo><mi>f</mi><mo>=</mo><mi>a</mi></mstyle></math>. radici <lb/>
minori advent-<lb/>
itiæ propositæ.
</s>
<lb/>
<s xml:id="echoid-s166" xml:space="preserve">
ex <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> nihil fit <lb/>
qui <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>&gt;</mo><mi>r</mi></mstyle></math> (c.3.)
[<emph style="it">Note: 
Sheet c.3 is Add MS 6782, 415.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s167" xml:space="preserve">
radici maiori.
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s168" xml:space="preserve">
radici minori.
</s>
<lb/>
<s xml:id="echoid-s169" xml:space="preserve">
radici maiori.
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s170" xml:space="preserve">
Notandum quod radices <lb/>
affirmatæ sunt illæ <lb/>
quæ quæsuntur <lb/>
et quæ <emph style="super">hic</emph> sunt negativæ, sunt coniugates affir-<lb/>
mativæ.
</s>
</p>
<pb file="add_6783_f065v" o="65v" n="130"/>
<pb file="add_6783_f066" o="66" n="131"/>
<pb file="add_6783_f066v" o="66v" n="132"/>
<pb file="add_6783_f067" o="67" n="133"/>
<head xml:id="echoid-head25" xml:space="preserve" xml:lang="lat">
e. De resolutione æquationum per reductione
</head>
<p xml:lang="lat">
<s xml:id="echoid-s171" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda.
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s172" xml:space="preserve">
sint antithesis
</s>
<lb/>
<s xml:id="echoid-s173" xml:space="preserve">
fiat solutio per antecedentia
</s>
<lb/>
<s xml:id="echoid-s174" xml:space="preserve">
et soluitur æquatio adventitia proposita.
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s175" xml:space="preserve">
æquatio hyperbolica, quæ <lb/>
nunquam habet coniugatum.
</s>
<lb/>
<s xml:id="echoid-s176" xml:space="preserve">
Neque igitur æquatio proposita.
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s177" xml:space="preserve">
æquatio eliptica, quæ semper <lb/>
habet coniugatum.
</s>
<s xml:id="echoid-s178" xml:space="preserve">
Et tum <lb/>
æquatio adventita proposita <lb/>
habet etiam coniugatam: <lb/>
nimirum:
</s>
<lb/>
<s xml:id="echoid-s179" xml:space="preserve">
Quæ suo loco reducetur.
</s>
</p>
<pb file="add_6783_f067v" o="67v" n="134"/>
<pb file="add_6783_f068" o="68" n="135"/>
<pb file="add_6783_f068v" o="68v" n="136"/>
<pb file="add_6783_f069" o="69" n="137"/>
<pb file="add_6783_f069v" o="69v" n="138"/>
<pb file="add_6783_f070" o="70" n="139"/>
<pb file="add_6783_f070v" o="70v" n="140"/>
<pb file="add_6783_f071" o="71" n="141"/>
<pb file="add_6783_f071v" o="71v" n="142"/>
<pb file="add_6783_f072" o="72" n="143"/>
<div xml:id="echoid-div39" type="page_commentary" level="2" n="39">
<p>
<s xml:id="echoid-s180" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s180" xml:space="preserve">
Against two of the equations on this page there are references to Stevin.
These equations, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mn>0</mn><mo>=</mo><mo>-</mo><mn>6</mn><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>0</mn><mo>=</mo><mo>+</mo><mn>6</mn><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> were treated by Stevin in
<emph style="it">L'arithmétique ... aussi l'algebre</emph> (1585)
in his discussion of cubic equation without a square term (Problem LXIX).
These equations were Stevin's first and second cases, treated at length on pages 305 to 311 and 311 to 314 respectively.
Although Stevin gave the roots of each equation in the cube root form that Harriot gives here,
he did not reduce them to their real equivalents.
Thus, even although he observed that 4 is a root of the first equation, he did not explicitly identify
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mo maxsize="1">[</mo></msqrt><mn>3</mn><mo maxsize="1">]</mo><mrow><mn>2</mn><mn>0</mn><mo>+</mo><msqrt><mrow><mn>3</mn><mn>9</mn><mn>2</mn></mrow></msqrt></mrow><mo>+</mo><msqrt><mo maxsize="1">[</mo></msqrt><mn>3</mn><mo maxsize="1">]</mo><mrow><mn>2</mn><mn>0</mn><mo>-</mo><msqrt><mrow><mn>3</mn><mn>9</mn><mn>2</mn></mrow></msqrt></mrow></mstyle></math> with 4. <lb/>
Harriot's work is continued on the next few sheets, Add MS 6783, f. 73, f. 74, f. 75.
See also Add MS 6786, f. 401, where he calculates
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mo maxsize="1">[</mo></msqrt><mn>3</mn><mo maxsize="1">]</mo><mrow><mn>2</mn><mn>0</mn><mo>+</mo><msqrt><mrow><mn>3</mn><mn>9</mn><mn>2</mn></mrow></msqrt></mrow><mo>+</mo><msqrt><mo maxsize="1">[</mo></msqrt><mn>3</mn><mo maxsize="1">]</mo><mrow><mn>2</mn><mn>0</mn><mo>-</mo><msqrt><mrow><mn>3</mn><mn>9</mn><mn>2</mn></mrow></msqrt></mrow></mstyle></math> arithmetically.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head26" xml:space="preserve" xml:lang="lat">
Exempla quarundam æquationem cum solutionibus <lb/>
per radices cubicas binomiorum explicatas.
<lb/>[<emph style="it">tr: 
Examples of certain cubic equations with solutions in terms of roots of binomes
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s182" xml:space="preserve">
Bin. 1
<lb/>[<emph style="it">tr: 
A binome of the first kind
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s183" xml:space="preserve">
Bin. 2
<lb/>[<emph style="it">tr: 
A binome of the second kind
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s184" xml:space="preserve">
Bin. 4
<lb/>[<emph style="it">tr: 
A binome of the fourth kind
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s185" xml:space="preserve">
(Exemplum Stevini <lb/>
sed ab illo <lb/>
non perfecte solutum)
<lb/>[<emph style="it">tr: 
An example from Stevin, but not completely solved by him.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s186" xml:space="preserve">
Bin. 5
<lb/>[<emph style="it">tr: 
A binome of the fifth kind
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s187" xml:space="preserve">
(Exemplum Stevini <lb/>
sed ab illo <lb/>
non perfecte solutum)
<lb/>[<emph style="it">tr: 
An example from Stevin, but not completely solved by him.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s188" xml:space="preserve">
Bin. 3
<lb/>[<emph style="it">tr: 
A binome of the third kind
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s189" xml:space="preserve">
Bin. 6
<lb/>[<emph style="it">tr: 
A binome of the sixth kind
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f072v" o="72v" n="144"/>
<pb file="add_6783_f073" o="73" n="145"/>
<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">
A continuation of the work begun on Add MS 6783, f. 73.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head27" xml:space="preserve" xml:lang="lat">
Quomodo duæ sequentes æquationes soluuntur <lb/>
aliter quam supra.
<lb/>[<emph style="it">tr: 
How the two following equations may be solved otherwise than above.
</emph>]<lb/>
</head>
<pb file="add_6783_f073v" o="73v" n="146"/>
<pb file="add_6783_f074" o="74" n="147"/>
<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 continuation of the work begun on Add MS 6783, f. 73. <lb/>
The <emph style="it">via generali</emph>, or general way, of solving equations is Viète's numerical method.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s194" xml:space="preserve">
via generali
<lb/>[<emph style="it">tr: 
by the general way
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f074v" o="74v" n="148"/>
<pb file="add_6783_f075" o="75" n="149"/>
<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">
A continuation of the work begun on Add MS 6783, f. 73 and f. 74.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s197" xml:space="preserve">
Henricus <lb/>
princeps
<lb/>[<emph style="it">tr: 
Prince Henry
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f075v" o="75v" n="150"/>
<pb file="add_6783_f076" o="76" n="151"/>
<p xml:lang="lat">
<s xml:id="echoid-s198" xml:space="preserve">
Esto radius circuli unitas.
</s>
<lb/>
<s xml:id="echoid-s199" xml:space="preserve">
Si quatuor rectæ sint continue proportionales, quarum prima <lb/>
et maxima sit radius, fuerint quod tres secundæ æquales subtendenti <lb/>
120 gradus et quartæ proportionalium: erit secunda latus nonagenti <lb/>
ordinato eidem circuli inscriptam 3(1) – 1(3) æquantur subtensæ gradum <lb/>
vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>3</mn></mrow></msqrt></mstyle></math>.
</s>
<lb/>
<s xml:id="echoid-s200" xml:space="preserve">
2(1) + 1(2) æquantur radio + 1(3) <lb/>
1(1) erit latus quatuordecanguli.
</s>
<lb/>
<s xml:id="echoid-s201" xml:space="preserve">
5(1) – 5(3) + 1(5) æquatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>3</mn></mrow></msqrt></mstyle></math> <lb/>
1(1) erit latus quindecanguli.
</s>
<lb/>
<s xml:id="echoid-s202" xml:space="preserve">
Quærantur latera nonanguli, quatuordecanguli, qindecanguli.
</s>
<lb/>
<s xml:id="echoid-s203" xml:space="preserve">
Esto Cubus 198 + <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>4</mn><mn>0</mn><mn>2</mn><mn>0</mn><mn>4</mn></mrow></msqrt></mstyle></math> quæratur binomium latus cubicum <lb/>
huius binomii.
</s>
<lb/>
<s xml:id="echoid-s204" xml:space="preserve">
Datum quam libet cubum ita secare ut corpus truncatum com-<lb/>
prehendatur a 32 triangulis æquilateris et sex quadratis.
</s>
</p>
<pb file="add_6783_f076v" o="76v" n="152"/>
<pb file="add_6783_f077" o="77" n="153"/>
<div xml:id="echoid-div43" type="page_commentary" level="2" n="43">
<p>
<s xml:id="echoid-s205" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s205" xml:space="preserve">
The reference to Simon Stevin is to his
<emph style="it">L'arithmétique ... aussi l'algebre</emph> (1585).
The equation written here as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>7</mn><mi>a</mi><mo>=</mo><mn>6</mn></mstyle></math> appears in Stevin's discussion
of cubic equation lacking a square term (Problem LXIX).
This equation was Stevin's third case, treated at length on pages 314 to 318.
See Add MS 6783, f. 72 for Stevin's first and second cases. <lb/>
Stevin went on to treat cubics lacking a linear term in Problem LXX,
though not the specific example <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>7</mn><mi>a</mi><mi>a</mi><mo>=</mo><mn>3</mn><mn>6</mn></mstyle></math> treated here by Harriot.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head28" xml:space="preserve">
(9.
</head>
<p xml:lang="lat">
<s xml:id="echoid-s207" xml:space="preserve">
eliptica
<lb/>[<emph style="it">tr: 
elliptic
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s208" xml:space="preserve">
Reductiones in istis chartis <lb/>
habentur apud Simonem Stevinum ut alios <lb/>
quas adposui ad illarum easdem rationes <lb/>
intelligantur.
<lb/>[<emph style="it">tr: 
The reductions in these sheets are to be found in Simon Stevin and others,
which I have put so that the same reasons as theirs may be understood.
</emph>]<lb/>
</s>
<s xml:id="echoid-s209" xml:space="preserve">
Sed nostra forma.
<lb/>[<emph style="it">tr: 
But in my form.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f077v" o="77v" n="154"/>
<pb file="add_6783_f078" o="78" n="155"/>
<head xml:id="echoid-head29" xml:space="preserve">
(8.
</head>
<pb file="add_6783_f078v" o="78v" n="156"/>
<pb file="add_6783_f079" o="79" n="157"/>
<head xml:id="echoid-head30" xml:space="preserve">
(7.
</head>
<pb file="add_6783_f079v" o="79v" n="158"/>
<pb file="add_6783_f080" o="80" n="159"/>
<head xml:id="echoid-head31" xml:space="preserve">
(6.
</head>
<pb file="add_6783_f080v" o="80v" n="160"/>
<pb file="add_6783_f081" o="81" n="161"/>
<head xml:id="echoid-head32" xml:space="preserve">
(5.2<emph style="super">o</emph>
</head>
<pb file="add_6783_f081v" o="81v" n="162"/>
<pb file="add_6783_f082" o="82" n="163"/>
<head xml:id="echoid-head33" xml:space="preserve">
(5.
</head>
<pb file="add_6783_f082v" o="82v" n="164"/>
<pb file="add_6783_f083" o="83" n="165"/>
<head xml:id="echoid-head34" xml:space="preserve">
(4.
</head>
<pb file="add_6783_f083v" o="83v" n="166"/>
<pb file="add_6783_f084" o="84" n="167"/>
<head xml:id="echoid-head35" xml:space="preserve">
(3.2<emph style="super">o</emph>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s210" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mo>=</mo><mn>1</mn><mn>6</mn><mn>7</mn></mstyle></math> et amplius. non 168. <lb/>
rationalis. (probat)
<lb/>[<emph style="it">tr: 
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mo>=</mo><mn>1</mn><mn>6</mn><mn>7</mn></mstyle></math> or more, not as much as 168, rational (proved)
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s211" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi><mo>=</mo><mn>1</mn><mn>0</mn><mn>4</mn></mstyle></math> et amlius. <lb/>
non 105. <lb/>
rationalis. (probat)
<lb/>[<emph style="it">tr: 
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mo>=</mo><mn>1</mn><mn>0</mn><mn>4</mn></mstyle></math> or more, not as much as 105, rational (proved)
</emph>]<lb/>
</s>
</p>&gt;
<p xml:lang="lat">
<s xml:id="echoid-s212" xml:space="preserve">
A. Nota quod illa æquatione <lb/>
habet solummodo unam radicem <lb/>
etsi cubus e triente &amp;c. <lb/>
et triplum quadrati &amp;c. <lb/>
sit = solido dato <lb/>
sed cubus e radice <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> plani <lb/>
coefficientis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>&lt;</mo></mstyle></math> solido dato.
<lb/>[<emph style="it">tr: 
A. Note that this equation has only one root, although the cube of a third etc. and three times the square etc.
is the given solid, but the cube of the root of a third of the plane coefficient is less than the given solid.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f084v" o="84v" n="168"/>
<pb file="add_6783_f085" o="85" n="169"/>
<div xml:id="echoid-div44" type="page_commentary" level="2" n="44">
<p>
<s xml:id="echoid-s213" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s213" xml:space="preserve">
Note the calculation at the bottom of the page leading to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>+</mo><mn>1</mn><mo>=</mo><msqrt><mrow><mo>-</mo><mn>3</mn></mrow></msqrt></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mi>a</mi><mo>-</mo><mn>1</mn><mo>=</mo><msqrt><mrow><mo>-</mo><mn>3</mn></mrow></msqrt></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head36" xml:space="preserve">
(3.
</head>
<p xml:lang="lat">
<s xml:id="echoid-s215" xml:space="preserve">
hyperbolica
<lb/>[<emph style="it">tr: 
hyperbolic
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f085v" o="85v" n="170"/>
<pb file="add_6783_f086" o="86" n="171"/>
<head xml:id="echoid-head37" xml:space="preserve">
(2.
</head>
<pb file="add_6783_f086v" o="86v" n="172"/>
<pb file="add_6783_f087" o="87" n="173"/>
<head xml:id="echoid-head38" xml:space="preserve">
(1.
</head>
<pb file="add_6783_f087v" o="87v" n="174"/>
<pb file="add_6783_f088" o="88" n="175"/>
<head xml:id="echoid-head39" xml:space="preserve" xml:lang="lat">
De reductione aequationum
<lb/>[<emph style="it">tr: 
On the reduction of equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s216" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi></mstyle></math> invenitur altera <lb/>
pagina
<lb/>[<emph style="it">tr: 
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi></mstyle></math> is found in the other page.
</emph>]<lb/>
[<emph style="it">Note: 
The other page mentioned here is Add MS 6783, f. 89.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6783_f088v" o="88v" n="176"/>
<pb file="add_6783_f089" o="89" n="177"/>
<div xml:id="echoid-div45" type="page_commentary" level="2" n="45">
<p>
<s xml:id="echoid-s217" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s217" xml:space="preserve">This sheet refers to pages 353 and 410 of Stevin's
<emph style="it">L'arithmétique ... aussi l'algebre</emph> (1585).
The following question appears on both pages.
</s>
<lb/>
<s xml:id="echoid-s218" xml:space="preserve">
page 353:
</s>
<lb/>
<quote xml:lang="fra">
Il nous faut doncques trouuer deux nombres tells, que le quarré de la moitie du premier soit egal au second,
&amp; que la produict du premier par le second + 5 soit 36.
<lb/>[<emph style="it">tr: 
We must therefore find two numbers, such that the square of half the first is equal to the second,
amd the product of the first and the second, plus 5, is 36.
</emph>]<lb/>
</quote>
<lb/>
<s xml:id="echoid-s219" xml:space="preserve">
page 40:
</s>
<quote xml:lang="fra">
Question XI. <lb/>
Trouuons deux nombres tells, que le quarré de la moitie du premier soit egal au seconde
&amp; que le produict du premier par le second + 5, soit 36.
<lb/>[<emph style="it">tr: 
Question XI. <lb/>
Find two numbers, such that the square of half the first is equal to the second,
and the product of the first and the second, plus 5, is 36.
</emph>]<lb/>
</quote>
<lb/>
<s xml:id="echoid-s220" xml:space="preserve">
Stevin's question is posed as a specific numerical problem but, as so often in his study of Stevin,
Harriot examines the question in general terms. <lb/>
See also Add MS 6783, f. 88, f. 88v; see also Add MS 6783, f. 37, f. 42 for other problems of the same kind.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s222" xml:space="preserve">
De reductione aequationum
<lb/>[<emph style="it">tr: 
On the reduction of equations
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s223" xml:space="preserve">
Stevin <lb/>
pag. 353. 410.
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s224" xml:space="preserve">
et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f089v" o="89v" n="178"/>
<pb file="add_6783_f090" o="90" n="179"/>
<pb file="add_6783_f090v" o="90v" n="180"/>
<pb file="add_6783_f091" o="91" n="181"/>
<head xml:id="echoid-head40" xml:space="preserve" xml:lang="lat">
e. De resolutione æquationum per reductione
</head>
<p xml:lang="lat">
<s xml:id="echoid-s225" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda.
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s226" xml:space="preserve">
Hyperbolica <lb/>
æquatio.
</s>
<lb/>
<s xml:id="echoid-s227" xml:space="preserve">
fiat solutio per antecedentia
</s>
<lb/>
<s xml:id="echoid-s228" xml:space="preserve">
et soluitur æquatio adventitia proposita.
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s229" xml:space="preserve">
æquatio hyperbolica.
</s>
</p>
<pb file="add_6783_f091v" o="91v" n="182"/>
<pb file="add_6783_f092" o="92" n="183"/>
<pb file="add_6783_f092v" o="92v" n="184"/>
<pb file="add_6783_f093" o="93" n="185"/>
<pb file="add_6783_f093v" o="93v" n="186"/>
<pb file="add_6783_f094" o="94" n="187"/>
<head xml:id="echoid-head41" xml:space="preserve" xml:lang="lat">
4.) De resolutione per reductionem.
<lb/>[<emph style="it">tr: 
On solution by reduction
</emph>]<lb/>
</head>
<pb file="add_6783_f094v" o="94v" n="188"/>
<pb file="add_6783_f095" o="95" n="189"/>
<p>
<s xml:id="echoid-s230" xml:space="preserve">
Memento <lb/>
Jerusalem
</s>
</p>
<pb file="add_6783_f095v" o="95v" n="190"/>
<pb file="add_6783_f096" o="96" n="191"/>
<head xml:id="echoid-head42" xml:space="preserve">
3.)
</head>
<pb file="add_6783_f096v" o="96v" n="192"/>
<pb file="add_6783_f097" o="97" n="193"/>
<head xml:id="echoid-head43" xml:space="preserve">
2.)
</head>
<pb file="add_6783_f097v" o="97v" n="194"/>
<pb file="add_6783_f098" o="98" n="195"/>
<div xml:id="echoid-div46" type="page_commentary" level="2" n="46">
<p>
<s xml:id="echoid-s231" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s231" xml:space="preserve">
This page is the first of a 29-page section on cubic equations. <lb/>
For a note on the tranlsation of 'genera', 'species', 'differentia', see Add MS 6783, f. 99. <lb/>
The upper left part of this page shows all possible combinations of positive and negative terms
in linear, quadratic, cubic, quartic, and quintic equations. <lb/>
The upper right half of the page contains a more sophisticated table,
where the possibility of null terms is also taken into account.
The box containing the figure 4, for instance, tallies the cases of cubic equation,
excluding cases where all terms are negative.
Thus, there is just 1 case of cubic equation with only a cube term;
there are 3 cases with a cube and linear term;
there are 3 cases with a cube and square term;
and there are 7 cases with a cube square, and linear term.
These cases are listed explicitly on the following page, Add MS 6783, f. 99. <lb/>
The table in the lower half of the page collects and summarizes this information.
Thus the total number of cubics, under the heading (3), is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>×</mo><mn>1</mn><mo>+</mo><mn>2</mn><mo>×</mo><mn>3</mn><mo>+</mo><mn>1</mn><mo>×</mo><mn>7</mn><mo>=</mo><mn>1</mn><mn>4</mn></mstyle></math>. <lb/>
Harriot also notes the totals if cases where all the terms are negative are also counted. <lb/>
For some drafts and rough versions of this page, see Add MS 6783, f. 406v, f. 409v, f. 404.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head44" xml:space="preserve" xml:lang="lat">
e.1.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s233" xml:space="preserve">
Apparatus, ad genera, species et differentias <lb/>
æquationum adventiturum.
<lb/>[<emph style="it">tr: 
Preparation, on the degrees, types, and cases of the equations in question.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s234" xml:space="preserve">
Collectio <lb/>
et <lb/>
summaria <lb/>
numeratio.
<lb/>[<emph style="it">tr: 
A gathering
and summary of the enumeration.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s235" xml:space="preserve">
Si pure negativa <lb/>
ad numerentur: <lb/>
Progressio summarum <lb/>
erit:
<lb/>[<emph style="it">tr: 
If the purely negatives are counted, the progression of sums will be:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s236" xml:space="preserve">
Ratio tripla
<lb/>[<emph style="it">tr: 
Triple ratio
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s237" xml:space="preserve">
Unde <lb/>
&amp;c.
<lb/>[<emph style="it">tr: 
Whence, etc.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f098v" o="98v" n="196"/>
<pb file="add_6783_f099" o="99" n="197"/>
<div xml:id="echoid-div47" type="page_commentary" level="2" n="47">
<p>
<s xml:id="echoid-s238" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s238" xml:space="preserve">
On this page, Harriot begins to list all possible cases of
linear, quadratic, cubic, and quartic equations.
The list is completed on the following page, Add MS 6783, f. 100. <lb/>
From the way he has set out the list, we may translate his categories as follows:
<emph style="it">genera</emph> = degrees;
<emph style="it">species</emph> = types, according to the number of terms;
<emph style="it">differentia</emph> = individual cases. <lb/>
The enumerations of these categories for equations of each degree
are given in the table on the upper right in Add MS 6783, f. 99.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head45" xml:space="preserve" xml:lang="lat">
e.2.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s240" xml:space="preserve">
Ordinatio æquationum adventiturum per <lb/>
genera, species, et differentias.
<lb/>[<emph style="it">tr: 
An ordering of the equations in question by degrees, types, and cases.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f099v" o="99v" n="198"/>
<pb file="add_6783_f100" o="100" n="199"/>
<div xml:id="echoid-div48" type="page_commentary" level="2" n="48">
<p>
<s xml:id="echoid-s241" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s241" xml:space="preserve">
Here Harriot completes his listing of all possible cases of quartic equation,
begun on Add MS 6783, f. 99.
He also begins an enumeration of quintic equations but it does not go very far.
He has already calculated on Add MS 6783, f. 98 that the total number of cases
according to this scheme would be 146.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head46" xml:space="preserve" xml:lang="lat">
e.3.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s243" xml:space="preserve">
Ordinatio æquationum
<lb/>[<emph style="it">tr: 
An ordering of the equations
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f100v" o="100v" n="200"/>
<pb file="add_6783_f101" o="101" n="201"/>
<div xml:id="echoid-div49" type="page_commentary" level="2" n="49">
<p>
<s xml:id="echoid-s244" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s244" xml:space="preserve">
On this page Harriot proves the two identities <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>a</mi><mn>3</mn></msup></mrow><mo>+</mo><mn>3</mn><mi>a</mi><mi>r</mi><mi>q</mi><mo>=</mo><mrow><msup><mi>r</mi><mn>3</mn></msup></mrow><mo>-</mo><mrow><msup><mi>q</mi><mn>3</mn></msup></mrow><mi>w</mi><mi>h</mi><mi>e</mi><mi>n</mi></mstyle></math> a = r - q <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>,</mo></mstyle></math><lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>a</mi><mn>3</mn></msup></mrow><mo>-</mo><mn>3</mn><mi>a</mi><mi>r</mi><mi>q</mi><mo>=</mo><mrow><msup><mi>r</mi><mn>3</mn></msup></mrow><mo>+</mo><mrow><msup><mi>q</mi><mn>3</mn></msup></mrow><mi>w</mi><mi>h</mi><mi>e</mi><mi>n</mi></mstyle></math> a = r + q <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>.</mo></mstyle></math><lb/>
These give him immediate solutions for equations that take these particular forms.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head47" xml:space="preserve" xml:lang="lat">
e.4.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s246" xml:space="preserve">
De duabus alijs æquationibus canonicis præter illas <lb/>
quæ sunt supra traditæ. (in chartis d.)
<lb/>[<emph style="it">tr: 
On two other canonical equations besides those that were treated above (in sheets lettered d).
</emph>]<lb/>
[<emph style="it">Note: 
Sheets paginated with the letter 'd' make up the section entitled 'De generatione æqutionum canonicarum'
in Add MS 6783, f. 183 to f. 163.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s247" xml:space="preserve">
Duæ species canonicæ <lb/>
sunt:
<lb/>[<emph style="it">tr: 
The two canonical forms are:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s248" xml:space="preserve">
Generatio 1<emph style="super">æ</emph>
<lb/>[<emph style="it">tr: 
Generation of the first
</emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s249" xml:space="preserve">
Et ita est. Est igitur.
<lb/>[<emph style="it">tr: 
And so it is. Therefore it is so.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s250" xml:space="preserve">
Aliter
<lb/>[<emph style="it">tr: 
Another way
</emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s251" xml:space="preserve">
Demonstratur ut supra.
<lb/>[<emph style="it">tr: 
Demonstrated as above
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s252" xml:space="preserve">
Generatio 2<emph style="super">æ</emph>
<lb/>[<emph style="it">tr: 
Generation of the second
</emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s253" xml:space="preserve">
Et ita est. Est igitur.
<lb/>[<emph style="it">tr: 
And so it is. Therefore it is so.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f101v" o="101v" n="202"/>
<pb file="add_6783_f102" o="102" n="203"/>
<div xml:id="echoid-div50" type="page_commentary" level="2" n="50">
<p>
<s xml:id="echoid-s254" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s254" xml:space="preserve">
Here Harriot investigates the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>c</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>a</mi></mstyle></math>.
From the identity proved on the previous page, Add MS 6783, f. 101, he knows that a solution is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi><mo>-</mo><mi>r</mi></mstyle></math>,
where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi><mi>r</mi><mo>=</mo><mi>b</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi><mi>q</mi><mi>q</mi><mo>-</mo><mi>r</mi><mi>r</mi><mi>r</mi><mo>=</mo><mn>2</mn><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>.
Solving these last two equations simultaneously, he obtains two values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>),
one of which is positive, the other negative.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head48" xml:space="preserve" xml:lang="lat">
e.5.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s256" xml:space="preserve">
Æquatio adventitia <lb/>
resolvenda.
<lb/>[<emph style="it">tr: 
The kind of equation to be solved
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s257" xml:space="preserve">
Canon ad inventionem.
<lb/>[<emph style="it">tr: 
Canon for finding the solution
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s258" xml:space="preserve">
Sed non est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi><mi>c</mi><mo>&gt;</mo><msqrt><mrow><mi>b</mi><mi>b</mi><mi>b</mi><mi>b</mi><mi>b</mi><mi>b</mi></mrow></msqrt></mstyle></math> &amp;c.
<lb/>[<emph style="it">tr: 
But it is not the case that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi><mi>c</mi><mo>&gt;</mo><msqrt><mrow><mi>b</mi><mi>b</mi><mi>b</mi><mi>b</mi><mi>b</mi><mi>b</mi></mrow></msqrt></mstyle></math> etc.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s259" xml:space="preserve">
Ergo non est radix 2<emph style="super">a</emph>. <lb/>
nisi negativa.
<lb/>[<emph style="it">tr: 
Therefore there is no second root unless it is negative.
</emph>]<lb/>
</s>
<s xml:id="echoid-s260" xml:space="preserve">
Tum erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mo>-</mo><mi>r</mi><mi>r</mi><mi>r</mi></mstyle></math> <lb/>
ut infra apparebit.
<lb/>[<emph style="it">tr: 
Then it will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mi>r</mi><mi>r</mi><mi>r</mi></mstyle></math>, as will appear below.
</emph>]<lb/>
</s>
<s xml:id="echoid-s261" xml:space="preserve">
Attamen.
<lb/>[<emph style="it">tr: 
Nevertheless.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s262" xml:space="preserve">
Hinc apparet quod <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mi>r</mi><mi>r</mi><mi>r</mi></mstyle></math> <lb/>
est æquale radici 2<emph style="it">æ</emph> <lb/>
supra.
<lb/>[<emph style="it">tr: 
Here it is clear that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mi>r</mi><mi>r</mi><mi>r</mi></mstyle></math> is equal to the second root above.
</emph>]<lb/>
</s>
<s xml:id="echoid-s263" xml:space="preserve">
Et tota solutio <lb/>
potuit fieri ex illis <lb/>
duabus radicibus.
<lb/>[<emph style="it">tr: 
And the full solution can be formed from these two roots.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s264" xml:space="preserve">
Ergo species resolutione inventa, erit:
<lb/>[<emph style="it">tr: 
Therefore the form of the discovered solution will be
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f102v" o="102v" n="204"/>
<pb file="add_6783_f103" o="103" n="205"/>
<div xml:id="echoid-div51" type="page_commentary" level="2" n="51">
<p>
<s xml:id="echoid-s265" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s265" xml:space="preserve">
Here Harriot investigates the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>c</mi><mi>c</mi><mo>=</mo><mo>-</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>.
From the identity proved on Add MS 6783, f. 101, he knows that a solution is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi><mo>+</mo><mi>r</mi></mstyle></math>,
where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi><mi>r</mi><mo>=</mo><mi>b</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi><mi>q</mi><mi>q</mi><mo>+</mo><mi>r</mi><mi>r</mi><mi>r</mi><mo>=</mo><mn>2</mn><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>.
Solving these last two equations simultaneously, he obtains values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>,
leading to the standard solution formula for this type of equation.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head49" xml:space="preserve" xml:lang="lat">
e.6.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s267" xml:space="preserve">
Æquatio adventitia <lb/>
resolvenda.
<lb/>[<emph style="it">tr: 
The kind of equation to be solved
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s268" xml:space="preserve">
Canon ad inventionem.
<lb/>[<emph style="it">tr: 
Canon for finding the solution
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s269" xml:space="preserve">
Harum duarum radicum <emph style="super">quadraticarum</emph> <lb/>
summa est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
The sum of these two quadratic roots is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s270" xml:space="preserve">
Una <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>q</mi><mi>q</mi><mi>q</mi></mstyle></math> <lb/>
Altera <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>r</mi><mi>r</mi><mi>r</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
One is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi><mi>q</mi><mi>q</mi></mstyle></math>, the other is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>r</mi><mi>r</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s271" xml:space="preserve">
Unde solutio potest fieri <lb/>
per istas duas radices; <lb/>
et ita fit ut infra.
<lb/>[<emph style="it">tr: 
Whence the solution can be formed from these two roots, thus as is done below.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s272" xml:space="preserve">
Attamen.
<lb/>[<emph style="it">tr: 
Nevertheless.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s273" xml:space="preserve">
Et inde, omnia erunt ut supra: <lb/>
ergo: si prima radix <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>q</mi></mstyle></math> <lb/>
erit: secunda <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>r</mi></mstyle></math> <lb/>
vel: una <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>q</mi></mstyle></math>. et altera <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>r</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
And from here on, all will be as above, <lb/>
therefore, if the first root is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math>, the second will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>, or one is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math>, the other <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s274" xml:space="preserve">
Ergo species resolutione inventa, erit:
<lb/>[<emph style="it">tr: 
Therefore the form of the discovered solution will be
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f103v" o="103v" n="206"/>
<pb file="add_6783_f104" o="104" n="207"/>
<div xml:id="echoid-div52" type="page_commentary" level="2" n="52">
<p>
<s xml:id="echoid-s275" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s275" xml:space="preserve">
Here Harriot solves the same equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>c</mi><mi>c</mi><mo>=</mo><mo>-</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> as on the previous page,
Add MS 6783, f. 103, using slightly different manipulations. <lb/>
For a rough version of this page see Add MS 6783, f. 402.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head50" xml:space="preserve" xml:lang="lat">
e.7.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s277" xml:space="preserve">
Æquatio adventitia <lb/>
resolvenda.
<lb/>[<emph style="it">tr: 
The kind of equation to be solved
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s278" xml:space="preserve">
Canon ad inventionem.
<lb/>[<emph style="it">tr: 
Canon for finding the solution
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s279" xml:space="preserve">
Ex æquatione adventitia quærendum quid <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>q</mi><mo>+</mo><mi>r</mi><mo>=</mo><mi>a</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
From this kind of equation, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi><mo>+</mo><mi>r</mi></mstyle></math> is to be sought, which is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s280" xml:space="preserve">
1<emph style="it">o</emph>, quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
First <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> is sought.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s281" xml:space="preserve">
Sed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>=</mo><msqrt><mo maxsize="1">[</mo></msqrt><mn>3</mn><mo maxsize="1">]</mo><mrow><mn>2</mn><mi>c</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>q</mi><mi>q</mi><mi>q</mi></mrow></mstyle></math>.
<lb/>[<emph style="it">tr: 
But <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>=</mo><msqrt><mo maxsize="1">[</mo></msqrt><mn>3</mn><mo maxsize="1">]</mo><mrow><mn>2</mn><mi>c</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>q</mi><mi>q</mi><mi>q</mi></mrow></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s282" xml:space="preserve">
Ergo per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math> 1<emph style="super">o</emph>.
<lb/>[<emph style="it">tr: 
Therefore from the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s283" xml:space="preserve">
idem quod: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math> 2<emph style="super">o</emph>.
<lb/>[<emph style="it">tr: 
the same as the second <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s284" xml:space="preserve">
<lb/>[...]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s285" xml:space="preserve">
Sed etiam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>=</mo><mfrac><mrow><mi>b</mi><mi>b</mi></mrow><mrow><mi>q</mi></mrow></mfrac></mstyle></math>.
<lb/>[<emph style="it">tr: 
But also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>=</mo><mfrac><mrow><mi>b</mi><mi>b</mi></mrow><mrow><mi>q</mi></mrow></mfrac></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s286" xml:space="preserve">
Ergo per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math> 1<emph style="super">o</emph>.
<lb/>[<emph style="it">tr: 
Therefore from the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s287" xml:space="preserve">
Ergo per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math> 2<emph style="super">o</emph>.
<lb/>[<emph style="it">tr: 
Therefore from the second <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s288" xml:space="preserve">
Ergo <emph style="st">fit</emph> fiunt species reso-<lb/>
lutionis.
<lb/>[<emph style="it">tr: 
Hence arise the forms of the solution.
</emph>]<lb/>
</s>
<s xml:id="echoid-s289" xml:space="preserve">
Et eadem <emph style="st">ut</emph> ut in chartis superiori.
<lb/>[<emph style="it">tr: 
And they are the same as in the sheet above.
</emph>]<lb/>
[<emph style="it">Note: 
The 'sheet above' is sheet e.6, that is, Add MS 6783, f. 103.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6783_f104v" o="104v" n="208"/>
<pb file="add_6783_f105" o="105" n="209"/>
<div xml:id="echoid-div53" type="page_commentary" level="2" n="53">
<p>
<s xml:id="echoid-s290" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s290" xml:space="preserve">
Here Harriot solves the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>c</mi><mi>c</mi><mo>=</mo><mo>-</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> already solved on Add MS 6783, f. 103 and f. 104,
manipulating the letters in a third different way to arrive at the same solution. <lb/>
At the end of this page, Harriot declares that the form of solution found by these methods are more useful than
those found from the generation of canonical equations in Section d.
In subsequent pages he continues to explore both forms.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head51" xml:space="preserve" xml:lang="lat">
e.7.2<emph style="super">o</emph>) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s292" xml:space="preserve">
Aliter.
<lb/>[<emph style="it">tr: 
Another way.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s293" xml:space="preserve">
Æquatio adventitia <lb/>
resolvenda.
<lb/>[<emph style="it">tr: 
The kind of equation to be solved
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s294" xml:space="preserve">
Canon.
<lb/>[<emph style="it">tr: 
Canon
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s295" xml:space="preserve">
Ergo una radix <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>q</mi><mi>q</mi><mi>q</mi></mstyle></math> <lb/>
et altera <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>r</mi><mi>r</mi><mi>r</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
Therefore one root is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi><mi>q</mi><mi>q</mi></mstyle></math> and the other is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>r</mi><mi>r</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s296" xml:space="preserve">
Ergo, species resolutionis inventa, erit: (ut supra)
<lb/>[<emph style="it">tr: 
Therefore the form of the solution is found, and will be (as above):
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s297" xml:space="preserve">
Hic obiter notandum, quod: <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>,</mo><mi>b</mi><mi>b</mi><mo>,</mo><mi>b</mi><mi>b</mi></mstyle></math> apte enuntiatur, plani solidum. vel quadrati cubus <lb/>
non autem cubo-cubus.
<lb/>[<emph style="it">tr: 
Here it is to be noted in passing that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>,</mo><mi>b</mi><mi>b</mi><mo>,</mo><mi>b</mi><mi>b</mi></mstyle></math> is rightly represented by
a solid formed from planes, or the cube of squares, but not the cube of a cube.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s298" xml:space="preserve">
Ita: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi><mi>c</mi><mo>,</mo><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math> enunciandum, solidi planum vel cubi <emph style="st">planum</emph> <emph style="super">quadratum</emph> <lb/>
non autem cubo-cubus.
<lb/>[<emph style="it">tr: 
Thus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi><mi>c</mi><mo>,</mo><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math> must be represented as
a plane formed from solids, or a square formed from cubes, but not the cube of a cube.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s299" xml:space="preserve">
Nam illæ enuntiationes <emph style="super">specierum</emph>
sunt magis commodæ quæ congru<emph style="super">u</emph>nt <lb/>
formis ex generatione adeptis, quam aliter ante fictis, præsertim <lb/>
in canonibus.
<lb/>[<emph style="it">tr: 
Now these representations of the forms of solution are more useful than the corresponding forms,
which were earlier obtained otherwise, from the generation of canonicals instead.
</emph>]<lb/>
</s>
<s xml:id="echoid-s300" xml:space="preserve">
alia tamen enuntiationes locum suum habent quando <lb/>
erit opportunum.
<lb/>[<emph style="it">tr: 
Other representations, however, have their place when there is opportunity.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f105v" o="105v" n="210"/>
<pb file="add_6783_f106" o="106" n="211"/>
<div xml:id="echoid-div54" type="page_commentary" level="2" n="54">
<p>
<s xml:id="echoid-s301" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s301" xml:space="preserve">
Here Harriot examines three possible cases of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>c</mi><mi>c</mi><mo>=</mo><mo>-</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>,
when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>&gt;</mo><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>&lt;</mo><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mi>b</mi></mstyle></math>. He calls these
hyperbolic, elliptic, and parabolic, respectively.
(For a page where Harriot experimented with other names for these forms, see Add MS 6784, f. 416.) <lb/>
When <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>&gt;</mo><mi>b</mi></mstyle></math>, the equation has just one real root,
given by the solution formula that Harriot has derived in sheets e.4, e.6, e.7, e.7.2
(Add MS 6783, f. 101, f. 103, f. 104, f. 105).
When <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mi>b</mi></mstyle></math>, the equation has the double root <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mo>-</mo><mi>b</mi></mstyle></math> and the single root <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn><mi>b</mi></mstyle></math>.
When <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>&lt;</mo><mi>b</mi></mstyle></math>, the equation has three distinct real roots, one positive and two negative;
this is the 'impossible' case where the solution formula appears to give imaginary roots.
However, Harriot has already shown in sheet d.4 (Add MS 6783, f. 180)
that such equations have a real positive root. <lb/>
Now Harriot wishes to show that, conversely, the canonical forms derived in d.4 and e.4
do in fact have the defining properties of hyperbolic, elliptic, or parabolic equations.
To do this, he makes use of two simple lemmas proved on the back of this page (add MS 6783, f. 106v). <lb/>
The canonical equation derived in e.4 was <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>3</mn><mi>q</mi><mi>r</mi><mi>a</mi><mo>=</mo><mi>r</mi><mi>r</mi><mi>r</mi><mo>+</mo><mi>q</mi><mi>q</mi><mi>q</mi></mstyle></math>.
To show that this equation is hyperbolic, Harriot needs to prove that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mfrac><mrow><mi>q</mi><mi>q</mi><mi>q</mi><mo>+</mo><mi>r</mi><mi>r</mi><mi>r</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mrow><msup><mo maxsize="1">)</mo><mn>2</mn></msup></mrow><mo>&gt;</mo><mo maxsize="1">(</mo><mi>q</mi><mi>r</mi><mrow><msup><mo maxsize="1">)</mo><mn>3</mn></msup></mrow></mstyle></math>;
this he is able to do using his two lemmas. <lb/>
The elliptic and parabolic cases are proved in sheet e.9, that is, Add MS 6783, f. 107.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head52" xml:space="preserve" xml:lang="lat">
e.8.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s303" xml:space="preserve">
De tribus casibus antecendentis æquationis adventitæ <lb/>
Videlicet:
<lb/>[<emph style="it">tr: 
On the three preceding case of the equation in question, namely:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s304" xml:space="preserve">
Primus casus habet:
<lb/>[<emph style="it">tr: 
The first case is:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s305" xml:space="preserve">
Æquatio primi casus, appelletur Hyperbolica: <lb/>
secundi casus, Eliptica: Tertij casus, Parabolica.
<lb/>[<emph style="it">tr: 
An equation in the first case is called 'hyperbolic'; in the second case 'elliptic'; in the third case 'parabolic'.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s306" xml:space="preserve">
Illarum trium æquationum, sunt tres proprij canones.
<lb/>[<emph style="it">tr: 
For these three equations, there are three corresponding canonical forms
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s307" xml:space="preserve">
Canon hyperblicus.
<lb/>[<emph style="it">tr: 
The hyperbolic canonical form
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s308" xml:space="preserve">
Elipticus.
<lb/>[<emph style="it">tr: 
The elliptic form
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s309" xml:space="preserve">
Parabolicus.
<lb/>[<emph style="it">tr: 
The parabolic form
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s310" xml:space="preserve">
Canon Hyperbolicus generatur supra (e.4.) Elipticus (d.4.) <lb/>
Parabolicus fit ex Hyperbolico vel Eliptico, mutando <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
The hyperbolic canonical form is generated above in e.4; the elliptic in d.4;
the parabolic arises from the hperblic or elliptic by changing <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>q</mi></mstyle></math>.
</emph>]<lb/>
[<emph style="it">Note: 
Sheet e.4 is Add MS 6783, f. 101; sheet d.4 is Add MS 6783, f. 180.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s311" xml:space="preserve">
Iam demonstrandum est quod in tribus istis canonibus sunt <lb/>
proprietates trium casuum superiorum.
<lb/>[<emph style="it">tr: 
Now it must be shown that these three canonical forms have the properties of the above three cases.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s312" xml:space="preserve">
Sed præmittenda sunt duo lemmata.
<lb/>[<emph style="it">tr: 
But first there are set out two lemmas.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s313" xml:space="preserve">
1<emph style="super">m</emph> Si sint tres continue proportionales inæquales: summa <lb/>
e duabus extremis maior est bis media.
<lb/>[<emph style="it">tr: 
1. If there are three unequal continued proportionals, the sum of the two extremes is greater than twice the mean.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s314" xml:space="preserve">
2<emph style="super">m</emph> Si sint quatour continue proportionales: summa e duabus <lb/>
extremis, maior est summa e duabus medijs.
<lb/>[<emph style="it">tr: 
2. If there are four continued proportinals, the sum of the two extremes is greater than the sum of the two means.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s315" xml:space="preserve">
Tum 1<emph style="super">o</emph> dico quod:
<lb/>[<emph style="it">tr: 
Then first, I say that:
</emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s316" xml:space="preserve">
Et ita est per 1<emph style="super">m</emph> lemma. Est igitur.
<lb/>[<emph style="it">tr: 
And it is so by the first lemma. Therefore it is so.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s317" xml:space="preserve">
Si: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>=</mo><mi>q</mi></mstyle></math>. <lb/>
erit æqualitas.
<lb/>[<emph style="it">tr: 
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>=</mo><mi>q</mi></mstyle></math> there will equality.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s318" xml:space="preserve">
Verte.
<lb/>[<emph style="it">tr: 
Turn over.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f106v" o="106v" n="212"/>
<div xml:id="echoid-div55" type="page_commentary" level="2" n="55">
<p>
<s xml:id="echoid-s319" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s319" xml:space="preserve">
Here Harriot proves two lemmas, needed to prove that the conical form derived in e.4
(Add MS 6783, f. 101) is hyperbolic.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s321" xml:space="preserve">
Lemma 1<emph style="super">m</emph>
<lb/>[<emph style="it">tr: 
Lemma 1.
</emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s322" xml:space="preserve">
Et ita est. Est igitur.
<lb/>[<emph style="it">tr: 
And so it is. Therefore it is so.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s323" xml:space="preserve">
Lemma 2<emph style="super">m</emph>. potest demonstravi eodem modo, Attamen
<lb/>[<emph style="it">tr: 
Lemma 2 may be demonstrated by the same method; nevertheless
</emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s324" xml:space="preserve">
quod fiat demonstrandum
<lb/>[<emph style="it">tr: 
which was to be proved
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f107" o="107" n="213"/>
<div xml:id="echoid-div56" type="page_commentary" level="2" n="56">
<p>
<s xml:id="echoid-s325" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s325" xml:space="preserve">
The canonical equation derived in d.4 (Add MS 6783, f. 180) was <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>q</mi><mi>q</mi><mi>a</mi><mo>-</mo><mi>q</mi><mi>r</mi><mi>a</mi><mo>-</mo><mi>r</mi><mi>r</mi><mi>a</mi><mo>=</mo><mi>q</mi><mi>q</mi><mi>r</mi><mo>+</mo><mi>q</mi><mi>r</mi><mi>r</mi></mstyle></math>.
To show that this equation is elliptic, Harriot needs to prove that
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mfrac><mrow><mi>q</mi><mi>q</mi><mi>r</mi><mo>+</mo><mi>q</mi><mi>r</mi><mi>r</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mrow><msup><mo maxsize="1">)</mo><mn>2</mn></msup></mrow><mo>&lt;</mo><mo maxsize="1">(</mo><mfrac><mrow><mi>q</mi><mi>q</mi><mo>+</mo><mi>q</mi><mi>r</mi><mo>+</mo><mi>r</mi><mi>r</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mrow><msup><mo maxsize="1">)</mo><mn>3</mn></msup></mrow></mstyle></math>, which he is able to do.
When <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>=</mo><mi>q</mi></mstyle></math>, we have equality, giving the parabolic case of the equation. <lb/>
Thus Harriot has proved what he set out to demonstrate,
as stated on the previous page, e.8 (Add MS 6783, f. 106).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head53" xml:space="preserve" xml:lang="lat">
e.9.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s327" xml:space="preserve">
Tum 2<emph style="super">o</emph> dico quod:
<lb/>[<emph style="it">tr: 
Then second, I say that:
</emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s328" xml:space="preserve">
Et ita est per lemmata. Est igitur.
<lb/>[<emph style="it">tr: 
And so it is by the lemmas. Therefore it is so.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s329" xml:space="preserve">
Si: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>=</mo><mi>q</mi></mstyle></math>. <lb/>
erit æqualitas.
<lb/>[<emph style="it">tr: 
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>=</mo><mi>q</mi></mstyle></math>, there will be equality.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s330" xml:space="preserve">
Tum 3<emph style="super">o</emph> dico quod:
<lb/>[<emph style="it">tr: 
Then third, I say that:
</emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s331" xml:space="preserve">
Et ita est. Est igitur.
<lb/>[<emph style="it">tr: 
And so it is. Therefore it is so.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f107v" o="107v" n="214"/>
<pb file="add_6783_f108" o="108" n="215"/>
<div xml:id="echoid-div57" type="page_commentary" level="2" n="57">
<p>
<s xml:id="echoid-s332" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s332" xml:space="preserve">
Up to now, Harriot has attempted to solve the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>c</mi><mi>c</mi><mo>=</mo><mo>-</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>
using the canonical form derived on e.4 (Add MS 6783, 101).
Now he tries using an alternative form, derived on d.4 (Add MS 6783, f. 180).
He must therefore set <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi><mi>q</mi><mo>+</mo><mi>q</mi><mi>r</mi><mo>+</mo><mi>r</mi><mi>r</mi><mo>=</mo><mn>3</mn><mi>b</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi><mi>q</mi><mi>r</mi><mo>+</mo><mi>q</mi><mi>r</mi><mi>r</mi><mo>=</mo><mn>2</mn><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>.
Unfortunately, attempting to solve these simultaneously only brings him back to the original equation,
except that the sign of the unknown quantity is now reversed;
Harriot calls this the conjugate equation (see Add MS 6783, f. 180).
He then tries using a second canonical form derived in d.4, but again is forced back to the original equation.
He concludes from this that the elliptic case can only be solved by means of its conjugate,
or else by the 'general way', that is, by Viète's numerical method.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head54" xml:space="preserve" xml:lang="lat">
e.10.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s334" xml:space="preserve">
Æquatio adventitia <lb/>
resolvenda.
<lb/>[<emph style="it">tr: 
The kind of equation to be solved
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s335" xml:space="preserve">
Canon.
<lb/>[<emph style="it">tr: 
Canon
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s336" xml:space="preserve">
Ex æquatione adventitia <lb/>
Quærendum quid <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi><mo>+</mo><mi>r</mi><mo>=</mo><mi>a</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
From this kind of equation, there must sought <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi><mo>+</mo><mi>r</mi></mstyle></math>, which is <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-s337" xml:space="preserve">
1<emph style="super">o</emph> quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
First <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> is sought.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s338" xml:space="preserve">
Et si quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math> ut antea, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
And if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math> is sought before <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s339" xml:space="preserve">
Erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>b</mi><mi>b</mi><mi>r</mi><mo>-</mo><mi>r</mi><mi>r</mi><mi>r</mi><mo>=</mo><mn>2</mn><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
We will have <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>b</mi><mi>b</mi><mi>r</mi><mo>-</mo><mi>r</mi><mi>r</mi><mi>r</mi><mo>=</mo><mn>2</mn><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s340" xml:space="preserve">
Utravis æquatio, adventitiæ <lb/>
coniugata.
<lb/>[<emph style="it">tr: 
Either equation is a conjugate of the equation in question.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s341" xml:space="preserve">
Aliter. per alium canonem ex (d.4.)
<lb/>[<emph style="it">tr: 
Another way, by another canon from d.4.
</emph>]<lb/>
[<emph style="it">Note: 
Sheet d.4 is Add MS 6783, f. 180. Here Harriot uses the second canon derived on that page.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s342" xml:space="preserve">
Quærendum ex adventitia quid <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>q</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
From the equation in question there must be sought <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s343" xml:space="preserve">
1<emph style="super">o</emph> quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
First <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> is sought.
</emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s344" xml:space="preserve">
Non opus est <lb/>
ulterius <lb/>
agere pro, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
There is no need to proceed further for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s345" xml:space="preserve">
Eadem æquatio ut <lb/>
adventitia.
<lb/>[<emph style="it">tr: 
The same as the equation in question.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s346" xml:space="preserve">
Illatum
<lb/>[<emph style="it">tr: 
An addition
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s347" xml:space="preserve">
Apparet igitur quod æquatio eliptica, videlicet 2<emph style="super">i</emph> casus, non <lb/>
soluitur per canones aliter, quam per coniugatam, vel immediate <lb/>
via generali.
<lb/>[<emph style="it">tr: 
It is clear, therefore, that an elliptic equation, namely the seocnd case,
cannot be solved by canons other than its conjugate, or in the general way.
</emph>]<lb/>
</s>
<s xml:id="echoid-s348" xml:space="preserve">
Coniugata etiam, ut post hæc apparebit non soluitur <lb/>
nisi eadem via generali.
<lb/>[<emph style="it">tr: 
Also the conjugate, as will afterwards appear, cannot be solved, except by the same general way.
</emph>]<lb/>
</s>
<s xml:id="echoid-s349" xml:space="preserve">
Quæ suo loco in antecedentibus <lb/>
fuit satis explicata.
<lb/>[<emph style="it">tr: 
Which, in its place in what has gone before, has been satisfactorily explained.
</emph>]<lb/>
</s>
<s xml:id="echoid-s350" xml:space="preserve">
Hoc tamen notandum, quod si <lb/>
Eliptica soluatur per suam coniugatam: summa e duabus <lb/>
radicibus coniugatæ, erit radix elipticæ quæsita.
<lb/>[<emph style="it">tr: 
However, this must be noted, that if an elliptic equation can be solved by its conjugate,
the sum of the two roots of the conjugate will be the sought root of the elliptic equation.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f108v" o="108v" n="216"/>
<pb file="add_6783_f109" o="109" n="217"/>
<div xml:id="echoid-div58" type="page_commentary" level="2" n="58">
<p>
<s xml:id="echoid-s351" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s351" xml:space="preserve">
Here Harriot solves the parabolic case of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>c</mi><mi>c</mi><mo>=</mo><mo>-</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, the special case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mi>c</mi></mstyle></math>.
It is obvious that the only positive solution is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn><mi>b</mi></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head55" xml:space="preserve" xml:lang="lat">
e.11.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s353" xml:space="preserve">
Æquatio adventitia <lb/>
resolvenda.
<lb/>[<emph style="it">tr: 
The kind of equation to be solved
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s354" xml:space="preserve">
Canon.
<lb/>[<emph style="it">tr: 
Canon
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s355" xml:space="preserve">
In hoc tertio casu ponitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
In this third case, suppose <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mi>b</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s356" xml:space="preserve">
Unde forma æquationis <lb/>
erit ut canonica.
<lb/>[<emph style="it">tr: 
Whence the form of the equation will be as the canonical form.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s357" xml:space="preserve">
Species igitur resolutionis est statim obvia; <lb/>
et per se patet.
<lb/>[<emph style="it">tr: 
The form of the solution is therefore immediately exposed; and is clear from this.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s358" xml:space="preserve">
forma Hyperbolica
<lb/>[<emph style="it">tr: 
hyperbolic form
</emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s359" xml:space="preserve">
Et aliter: sed non opus est.
<lb/>[<emph style="it">tr: 
And otherwise; but there is no need.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s360" xml:space="preserve">
Hactenus de tribus casibus <lb/>
æquationis adventitiæ:
<lb/>[<emph style="it">tr: 
Up to now we have dealt wtih three cases of the equation in question.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f109v" o="109v" n="218"/>
<pb file="add_6783_f110" o="110" n="219"/>
<div xml:id="echoid-div59" type="page_commentary" level="2" n="59">
<p>
<s xml:id="echoid-s361" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s361" xml:space="preserve">
Here Harriot turns to equations of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>c</mi><mi>c</mi><mo>=</mo><mo>+</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>.
These are the 'avulsed equations' treated numerically in Section c (Add MS 6782. f. 417 to f. 400.)
Such equations may have either two positive roots or none.
Harriot begins with a canonical form derived in sheet d.3 (Add MS 6783, f. 181) for an equaion with two positive roots.
He must therefore set <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi><mi>q</mi><mo>+</mo><mi>q</mi><mi>r</mi><mo>+</mo><mi>r</mi><mi>r</mi><mo>=</mo><mn>3</mn><mi>b</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi><mi>q</mi><mi>r</mi><mo>+</mo><mi>q</mi><mi>r</mi><mi>r</mi><mo>=</mo><mn>2</mn><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>.
However, this simply brings him back to the origina equaton.
He therefore tries another canon, from d.4 (Add MS 6783, f. 180),
which requires him to set <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi><mi>q</mi><mo>-</mo><mi>q</mi><mi>r</mi><mo>+</mo><mi>r</mi><mi>r</mi><mo>=</mo><mn>3</mn><mi>b</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi><mi>q</mi><mi>r</mi><mo>-</mo><mi>q</mi><mi>r</mi><mi>r</mi><mo>=</mo><mn>2</mn><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>.
This brings him back to the conjugate of the original equation.
He therefore concludes, as on sheet e.10 (Add MS 6783, f. 108)
that the given equation can only be solved by means of its conjugate,
or else by Viète's numerical method.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head56" xml:space="preserve" xml:lang="lat">
e.12.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s363" xml:space="preserve">
Æquatio adventitia <lb/>
resolvenda.
<lb/>[<emph style="it">tr: 
The kind of equation to be solved
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s364" xml:space="preserve">
Canon. ex (d.3.)
<lb/>[<emph style="it">tr: 
Canon, from d.3.
</emph>]<lb/>
[<emph style="it">Note: 
Sheet d.3. is Add MS 6783, f. 181.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s365" xml:space="preserve">
Ex æquatione adventitia <lb/>
Quærendum quid <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>q</mi><mo>=</mo><mi>a</mi></mstyle></math>. <lb/>
quid <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>r</mi><mo>=</mo><mi>a</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
From the equation in question there must be sought <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math>, or <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>r</mi></mstyle></math>, or <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-s366" xml:space="preserve">
1<emph style="super">o</emph> quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
First, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> is sought.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s367" xml:space="preserve">
Utraque æquationes eædem <lb/>
ut adventitia.
<lb/>[<emph style="it">tr: 
Either equation is the same as that in question.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s368" xml:space="preserve">
Unde apparet quod adventitia <lb/>
habet necessario duas radices <lb/>
ut canon, hoc est
<lb/>[<emph style="it">tr: 
Whence it appears that the equation in question must have two roots, as does the canon, that is
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s369" xml:space="preserve">
Aliter. per alium canonem ex (d.4.)
<lb/>[<emph style="it">tr: 
Otherwise, by another canon from d.4.
</emph>]<lb/>
[<emph style="it">Note: 
Sheet d.4 is Add MS 6783, f. 180.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s370" xml:space="preserve">
Quærendum ex adventitia quid <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>q</mi><mo>-</mo><mi>r</mi><mo>=</mo><mi>a</mi></mstyle></math>. <lb/>
quid <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>r</mi><mo>=</mo><mi>a</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
From the equation in question there must be sought <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi><mo>-</mo><mi>r</mi></mstyle></math>, which 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>r</mi></mstyle></math>, which is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s371" xml:space="preserve">
1<emph style="super">o</emph> quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
First, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> is sought.
</emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s372" xml:space="preserve">
&amp;c. ut (e.10.)
<lb/>[<emph style="it">tr: 
etc. as in e.10.
</emph>]<lb/>
[<emph style="it">Note: 
Sheet e.10 is Add MS 6783, f. 108.
 </emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s373" xml:space="preserve">
æquatio adventitia coniugatæ <lb/>
unde <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math> est determinata et æqualis <lb/>
summa radicum ex adventitia.
<lb/>[<emph style="it">tr: 
the conjugate of the equation in question,
whence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math> is determined and is equal to the sum of the roots of the equation in question
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s374" xml:space="preserve">
Illatum de resolutione.
<lb/>[<emph style="it">tr: 
An additional note on the solution
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s375" xml:space="preserve">
Hinc apparet quod hæc æquatio adventitia <lb/>
non soluitur per canones aliter quam <lb/>
per suam coniugatum.
<lb/>[<emph style="it">tr: 
Here is is clear that the equation in question cannot be solved by canonical forms other than by its conjugate.
</emph>]<lb/>
</s>
<s xml:id="echoid-s376" xml:space="preserve">
Vel immediate <emph style="st">per</emph> <lb/>
via generali. &amp;c. ut est supra dictum.
<lb/>[<emph style="it">tr: 
Or completely in the general way, etc. as was said above.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f110v" o="110v" n="220"/>
<pb file="add_6783_f111" o="111" n="221"/>
<div xml:id="echoid-div60" type="page_commentary" level="2" n="60">
<p>
<s xml:id="echoid-s377" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s377" xml:space="preserve">
Here Harriot shows how the roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> of an equation of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>g</mi><mi>h</mi><mo>=</mo><mi>d</mi><mi>f</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>
can be determined from the known root <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi><mo>+</mo><mi>r</mi><mo>=</mo><mi>x</mi></mstyle></math> of the conjugate equation <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>f</mi><mi>a</mi><mo>=</mo><mi>g</mi><mi>g</mi><mi>h</mi></mstyle></math>.
To do this, he uses the first canonical form found on d.4 (Add MS 6783, f. 180).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head57" xml:space="preserve" xml:lang="lat">
e.13.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s379" xml:space="preserve">
Solutio antecedentis adventitiæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>g</mi><mi>h</mi><mo>=</mo><mi>d</mi><mi>f</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> <lb/>
ex data solutione coniugatæ.
<lb/>[<emph style="it">tr: 
The solution of the foregoing equation in question <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>g</mi><mi>h</mi><mo>=</mo><mi>d</mi><mi>f</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> from a given solution of its conjugate.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s380" xml:space="preserve">
Coniugata est <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>f</mi><mi>a</mi><mo>=</mo><mi>g</mi><mi>g</mi><mi>h</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
The conjugate is <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>f</mi><mi>a</mi><mo>=</mo><mi>g</mi><mi>g</mi><mi>h</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s381" xml:space="preserve">
Sit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>x</mi><mo>=</mo><mi>q</mi><mo>+</mo><mi>r</mi></mstyle></math> in canone 1<emph style="super">o</emph>.
<lb/>[<emph style="it">tr: 
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>x</mi><mo>=</mo><mi>q</mi><mo>+</mo><mi>r</mi></mstyle></math> in the first canonical form.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s382" xml:space="preserve">
1<emph style="super">o</emph> Inveniatur species resolutionis <lb/>
per primum canonem adventitiæ.
<lb/>[<emph style="it">tr: 
First, there is to be found the form of the solution from the first canon for the equation in question.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s383" xml:space="preserve">
Sit una radix quæsita <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mi>q</mi></mstyle></math>. <lb/>
Erit altera: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mo>-</mo><mi>e</mi><mo>=</mo><mi>r</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
Let one of the sought roots be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mi>q</mi></mstyle></math>, then the other will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mo>-</mo><mi>e</mi><mo>=</mo><mi>r</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s384" xml:space="preserve">
Summa duarum istarum radicum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>x</mi></mstyle></math> <lb/>
una <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>e</mi></mstyle></math>. altera <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>x</mi><mo>-</mo><mi>e</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
The sum of these two roots is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math>; one is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, the other is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mo>-</mo><mi>e</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s385" xml:space="preserve">
Ergo duæ istæ radices sunt <lb/>
species resolutionis quæsitæ.
<lb/>[<emph style="it">tr: 
Therefore these two roots are the sought form of the solution.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s386" xml:space="preserve">
(factum duarum <lb/>
radicum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>x</mi><mi>x</mi><mo>-</mo><mi>d</mi><mi>f</mi></mstyle></math>.)
<lb/>[<emph style="it">tr: 
(The product of the two roots is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mo>-</mo><mi>d</mi><mi>f</mi></mstyle></math>.)
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s387" xml:space="preserve">
Aliter per primum canone.
<lb/>[<emph style="it">tr: 
Another way by the firts canon.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s388" xml:space="preserve">
Sit una radix quæsita <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mi>q</mi></mstyle></math>. <lb/>
Erit altera: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mo>-</mo><mi>e</mi><mo>=</mo><mi>r</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
Let one of the roots sought be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mi>q</mi></mstyle></math>, then the other will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mo>-</mo><mi>e</mi><mo>=</mo><mi>r</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s389" xml:space="preserve">
Summa duarum istarum radicum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>x</mi></mstyle></math> <lb/>
una <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>e</mi></mstyle></math>. altera <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>x</mi><mo>-</mo><mi>e</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
The sum of these two roots is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math>; one is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, the other <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mo>-</mo><mi>e</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s390" xml:space="preserve">
Ergo duæ istæ radices sunt <lb/>
etiam species resolutionis quæsitæ.
<lb/>[<emph style="it">tr: 
Therefore these two roots are also the sought form of the solution.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s391" xml:space="preserve">
Sed priores species sunt ad reso-<lb/>
lutionem magis ad commodæ.
<lb/>[<emph style="it">tr: 
But the previous form is more useful for solution.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f111v" o="111v" n="222"/>
<pb file="add_6783_f112" o="112" n="223"/>
<div xml:id="echoid-div61" type="page_commentary" level="2" n="61">
<p>
<s xml:id="echoid-s392" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s392" xml:space="preserve">
As on sheet e.13 (Add MS 6783, f. 111) Harriot shows how the roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>
of an equation of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>g</mi><mi>h</mi><mo>=</mo><mi>d</mi><mi>f</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>
can be determined from a known root <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math> of the conjugate equation <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>f</mi><mi>a</mi><mo>=</mo><mi>g</mi><mi>g</mi><mi>h</mi></mstyle></math>. <lb/>
His second method uses the second canonical form from d.4 (Add MS 6783, f. 180). <lb/>
His third method uses straightforward algebraic manipulation to arrive at the same result. <lb/>
At the end of the sheet is a summary of what has been treated so far in Section e:
the three equations <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>g</mi><mi>h</mi><mo>=</mo><mo>+</mo><mi>d</mi><mi>f</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>g</mi><mi>g</mi><mi>h</mi><mo>=</mo><mo>-</mo><mi>d</mi><mi>f</mi><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>g</mi><mi>h</mi><mo>=</mo><mo>+</mo><mi>d</mi><mi>f</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>,
but not linear or quadratic equations, which Harriot regards as already sufficently well understood.
In the remainder of this section, he will show how to reduce any other cubic equation to one of these three types,
by removing the square term. This is the 'reduction' that appears in the title throughout this section.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head58" xml:space="preserve" xml:lang="lat">
e.14.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s394" xml:space="preserve">
2<emph style="super">o</emph> per 2<emph style="it">um</emph> canonem adventitiæ.
<lb/>[<emph style="it">tr: 
Second, by the second canon for the equation in question.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s395" xml:space="preserve">
In 2<emph style="super">o</emph> canone <emph style="super">erit</emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mo>=</mo><mi>q</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
In the second canon, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mo>=</mo><mi>q</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s396" xml:space="preserve">
Tum si una radix quæsita ponatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mi>r</mi></mstyle></math> <lb/>
altera erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mo>-</mo><mi>e</mi><mo>=</mo><mi>q</mi><mo>-</mo><mi>r</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
Then if one of the roots sought is supposed to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mi>r</mi></mstyle></math>, then the other will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mo>-</mo><mi>e</mi><mo>=</mo><mi>q</mi><mo>-</mo><mi>r</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s397" xml:space="preserve">
Tum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mo>-</mo><mi>x</mi><mi>e</mi><mo>+</mo><mi>e</mi><mi>e</mi><mo>=</mo><mi>d</mi><mi>f</mi></mstyle></math> <lb/>
&amp;c. ut in 1<emph style="super">o</emph> modo.
<lb/>[<emph style="it">tr: 
Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mo>-</mo><mi>x</mi><mi>e</mi><mo>+</mo><mi>e</mi><mi>e</mi><mo>=</mo><mi>d</mi><mi>f</mi></mstyle></math> etc., as in the first method.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s398" xml:space="preserve">
3<emph style="super">o</emph>. sine canone per poristicum (d.19.) <lb/>
ut Analystæ ante nos.
<lb/>[<emph style="it">tr: 
Third, without a canon, from the rule in sheet d.19, as analysts did before us.
</emph>]<lb/>
[<emph style="it">Note: 
Sheet d. 19 is Add MS 6783, f. 165.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s399" xml:space="preserve">
Adventitia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>f</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>g</mi><mi>g</mi><mi>h</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
The equation in question is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>f</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>g</mi><mi>g</mi><mi>h</mi></mstyle></math>.
</emph>]<lb/>
<lb/>
<lb/>[...]<lb/>
<lb/>
per poristicum (d.19.)
<lb/>[<emph style="it">tr: 
By the rule on sheet d.19.
</emph>]<lb/>
[<emph style="it">Note: 
Sheet d. 19 is Add MS 6783, f. 165.
 </emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s400" xml:space="preserve">
&amp;c. ut in 1<emph style="super">o</emph> modo.
<lb/>[<emph style="it">tr: 
etc. as in the first method.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s401" xml:space="preserve">
Hactenus <lb/>
de tribus primis binomijs <lb/>
æquationibus cubicis <lb/>
et illorum solutione: <lb/>
Videlicet:
<lb/>[<emph style="it">tr: 
Up to now we have treated the first three binomial cases of cubic equation and their solutions, namely:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s402" xml:space="preserve">
omittuntur <lb/>
Harum derivativæ, <lb/>
Æquationes simplices, <lb/>
Et quadraticæ, <lb/>
ut satis notæ.
<lb/>[<emph style="it">tr: 
There are ommitted: their derivations, simple equations, and quadratic equations, as being sufficiently known.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s403" xml:space="preserve">
Expectandum <lb/>
in sequentibus; <lb/>
Solummodo reductio <lb/>
ad alias species <lb/>
antecedentas <lb/>
unde fiat <lb/>
solutio.
<lb/>[<emph style="it">tr: 
There is to be expected in what follows only the reduction to other foregoing types, whence will come the solution.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f112v" o="112v" n="224"/>
<pb file="add_6783_f113" o="113" n="225"/>
<div xml:id="echoid-div62" type="page_commentary" level="2" n="62">
<p>
<s xml:id="echoid-s404" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s404" xml:space="preserve">
This is the first page of a 7-page section on removing the cube term from a quartic equation;
this is the 'reduction' that appears in the headings throughout Section f.
Harriot's friend and colleague Nathaniel Torporley later denoted this subsection by the letter α
(see Stedall 2003, 247–258).
It is not self-contained, since the pagination (here 1 to 7) later continues into section γ (8 to 19)
(see Stedall 2003, 259–275),
but it is convenient to treat these first seven pages together.
There is also another version of it, for which Torporley used the letter β,
in Add MS 6783, f. 145 to f. 139. <lb/>
This first page shows the removal of the cube term from the quartic equations <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mo maxsize="1.01">/</mo><mi>p</mi><mi>m</mi><mn>4</mn><mi>r</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>,
using the substitution <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mo>±</mo><mi>e</mi><mo>∓</mo><mi>r</mi></mstyle></math>.
Harriot then solves the resulting equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, either by inspection or by the methods shown in subsection δ
(see Add MS 6783, f. 137). This gives him the two real roots of the original equation.
(The other two are imaginary.)
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head59" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (1.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s406" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f113v" o="113v" n="226"/>
<pb file="add_6783_f114" o="114" n="227"/>
<div xml:id="echoid-div63" type="page_commentary" level="2" n="63">
<p>
<s xml:id="echoid-s407" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s407" xml:space="preserve">
The method of solution here is the same as that on Add MS 6783, f. 113.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head60" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (2.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s409" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f114v" o="114v" n="228"/>
<pb file="add_6783_f115" o="115" n="229"/>
<div xml:id="echoid-div64" type="page_commentary" level="2" n="64">
<p>
<s xml:id="echoid-s410" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s410" xml:space="preserve">
The method of solution here is the same as that on Add MS 6783, f. 113.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head61" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (3.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s412" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f115v" o="115v" n="230"/>
<pb file="add_6783_f116" o="116" n="231"/>
<div xml:id="echoid-div65" type="page_commentary" level="2" n="65">
<p>
<s xml:id="echoid-s413" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s413" xml:space="preserve">
The method of solution here is the same as that on Add MS 6783, f. 113.
A numerical example of the method follows on Add MS 6783, f. 117.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head62" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (4.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s415" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f116v" o="116v" n="232"/>
<pb file="add_6783_f117" o="117" n="233"/>
<div xml:id="echoid-div66" type="page_commentary" level="2" n="66">
<p>
<s xml:id="echoid-s416" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s416" xml:space="preserve">
This page gives a numerical example of the method outlined on Add MS 6783, f. 116.
Here Harriot is able to solve each quartic equation for all four real roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head63" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (5.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s418" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f117v" o="117v" n="234"/>
<pb file="add_6783_f118" o="118" n="235"/>
<div xml:id="echoid-div67" type="page_commentary" level="2" n="67">
<p>
<s xml:id="echoid-s419" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s419" xml:space="preserve">
The method of solution here is the same as that on Add MS 6783, f. 113.
A numerical example of the method follows on Add MS 6783, f. 130,
which for some reason has become separated from the rest of this section.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head64" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (6.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s421" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f118v" o="118v" n="236"/>
<pb file="add_6783_f119" o="119" n="237"/>
<head xml:id="echoid-head65" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (7.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s422" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f119v" o="119v" n="238"/>
<div xml:id="echoid-div68" type="page_commentary" level="2" n="68">
<p>
<s xml:id="echoid-s423" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s423" xml:space="preserve">
Some quartic equations and their roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6783_f120" o="120" n="239"/>
<pb file="add_6783_f120v" o="120v" n="240"/>
<pb file="add_6783_f121" o="121" n="241"/>
<pb file="add_6783_f121v" o="121v" n="242"/>
<pb file="add_6783_f122" o="122" n="243"/>
<pb file="add_6783_f122v" o="122v" n="244"/>
<pb file="add_6783_f123" o="123" n="245"/>
<pb file="add_6783_f123v" o="123v" n="246"/>
<pb file="add_6783_f124" o="124" n="247"/>
<pb file="add_6783_f124v" o="124v" n="248"/>
<pb file="add_6783_f125" o="125" n="249"/>
<pb file="add_6783_f125v" o="125v" n="250"/>
<pb file="add_6783_f126" o="126" n="251"/>
<pb file="add_6783_f126v" o="126v" n="252"/>
<pb file="add_6783_f127" o="127" n="253"/>
<pb file="add_6783_f127v" o="127v" n="254"/>
<pb file="add_6783_f128" o="128" n="255"/>
<pb file="add_6783_f128v" o="128v" n="256"/>
<pb file="add_6783_f129" o="129" n="257"/>
<pb file="add_6783_f129v" o="129v" n="258"/>
<pb file="add_6783_f130" o="130" n="259"/>
<div xml:id="echoid-div69" type="page_commentary" level="2" n="69">
<p>
<s xml:id="echoid-s425" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s425" xml:space="preserve">
This page gives a numerical example of the method outlined on Add MS 6783, f. 118.
Here Harriot is able to solve each quartic equation for all four real roots.
Note his use of the possibility <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mn>0</mn></mstyle></math>, separated into the two cases <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mo>+</mo><mn>0</mn></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mo>-</mo><mn>0</mn></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head66" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (7.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s427" xml:space="preserve">
Antecedentes æquationes:
<lb/>[<emph style="it">tr: 
The preceding equations:
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f130v" o="130v" n="260"/>
<pb file="add_6783_f131" o="131" n="261"/>
<div xml:id="echoid-div70" type="page_commentary" level="2" n="70">
<p>
<s xml:id="echoid-s428" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s428" xml:space="preserve">
The method of solution here is essentially the same as that on Add MS 6783, f. 134.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head67" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s430" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f131v" o="131v" n="262"/>
<pb file="add_6783_f132" o="132" n="263"/>
<div xml:id="echoid-div71" type="page_commentary" level="2" n="71">
<p>
<s xml:id="echoid-s431" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s431" xml:space="preserve">
The method of solution here is essentially the same as that on Add MS 6783, f. 134.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head68" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s433" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f132v" o="132v" n="264"/>
<pb file="add_6783_f133" o="133" n="265"/>
<div xml:id="echoid-div72" type="page_commentary" level="2" n="72">
<p>
<s xml:id="echoid-s434" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s434" xml:space="preserve">
The method of solution here is essentially the same as that on Add MS 6783, f. 134.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head69" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s436" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f133v" o="133v" n="266"/>
<pb file="add_6783_f134" o="134" n="267"/>
<div xml:id="echoid-div73" type="page_commentary" level="2" n="73">
<p>
<s xml:id="echoid-s437" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s437" xml:space="preserve">
The method of solution here is essentially the same as that on Add MS 6783, f. 135. <lb/>
The resulting equation in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>e</mi></mstyle></math> is a full cubic equation.
Harriot makes the substitution <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>e</mi><mo>=</mo><mi>u</mi><mi>u</mi><mo>+</mo><mi>p</mi><mi>p</mi></mstyle></math> to eliminate the 'square' term.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head70" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s439" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f134v" o="134v" n="268"/>
<pb file="add_6783_f135" o="135" n="269"/>
<div xml:id="echoid-div74" type="page_commentary" level="2" n="74">
<p>
<s xml:id="echoid-s440" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s440" xml:space="preserve">
As in Add MS 6783, f. 137 and f. 138, these equations are solved by completing a square.
Harriot does so by adding a quantity <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>e</mi><mi>a</mi><mi>a</mi><mo>+</mo><mfrac><mrow><mi>e</mi><mi>e</mi><mi>e</mi><mi>e</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> to each side of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>=</mo><mo>∓</mo><mn>2</mn><mi>c</mi><mi>c</mi><mi>d</mi><mi>a</mi><mo>+</mo><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>.
The condition for the right-hand side to be a perfect square is that
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>e</mi><mo maxsize="1">(</mo><mfrac><mrow><mi>e</mi><mi>e</mi><mi>e</mi><mi>e</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo>+</mo><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo maxsize="1">)</mo><mo>=</mo><mo maxsize="1">(</mo><mi>c</mi><mi>c</mi><mi>d</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>c</mi><mi>c</mi><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>.
This gives him a cubic equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, which in principle can be solved by the methods demonstrated in Section e. <lb/>
This is the standard method already taught by Cardano, Bombelli, Stevin, and Viète for solving quartic equations.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head71" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s442" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s443" xml:space="preserve">
Datur igitur; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>e</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>e</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f135v" o="135v" n="270"/>
<pb file="add_6783_f136" o="136" n="271"/>
<div xml:id="echoid-div75" type="page_commentary" level="2" n="75">
<p>
<s xml:id="echoid-s444" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s444" xml:space="preserve">
The method of solution here is the same as that on Add MS 6783, f. 135.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head72" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s446" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s447" xml:space="preserve">
Datur igitur; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>e</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>e</mi></mstyle></math> is given.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f136v" o="136v" n="272"/>
<pb file="add_6783_f137" o="137" n="273"/>
<div xml:id="echoid-div76" type="page_commentary" level="2" n="76">
<p>
<s xml:id="echoid-s448" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s448" xml:space="preserve">
In a set of sheets lettered 'e' (Add MS 6783, f. 98 to f. 112, f. 216 to f. 219, and f. 198 to f. 184)
Harriot gave a comprehensive treatment of cubic equations.
In a similar set of sheets lettered 'f' he did the same for quartic equations.
Because there are many more cases of quartic to deal with, the sheets are more numerous,
and are more widely dispersed amongst the mansucripts.
After Harriot's death his colleague Nathaniel Torporley gave Greek letters to certain subsections of Section f,
and we have followed that convention here.
For further details of Torporley's notes on these pages, see Stedall 2003, 17–26. <lb/>
This is the first page of the 8-page section to which Torporley gave the letter δ (see Stedall 2003, 239–247).
The reason for putting this section first in the present scheme is that it treats quartic equations with no cube term,
which can therefore be solved by 'completing the square'.
The aim of all the other sections is to reduce more general quartic equations to this type
so that they can be solved in the same way.
This preserves an analogy with Harriot's treatment of cubics, where cubics with no square term were treated first,
and all other case were then reduced to this special type.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head73" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s450" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f137v" o="137v" n="274"/>
<pb file="add_6783_f138" o="138" n="275"/>
<div xml:id="echoid-div77" type="page_commentary" level="2" n="77">
<p>
<s xml:id="echoid-s451" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s451" xml:space="preserve">
As on Add MS 6783, f. 137, these equations are solved immediately by completing the square.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head74" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s453" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f138v" o="138v" n="276"/>
<pb file="add_6783_f139" o="139" n="277"/>
<div xml:id="echoid-div78" type="page_commentary" level="2" n="78">
<p>
<s xml:id="echoid-s454" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s454" xml:space="preserve">
The method of solution here is essentially the same as that on Add MS 6783, f. 145.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head75" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (7.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s456" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f139v" o="139v" n="278"/>
<pb file="add_6783_f140" o="140" n="279"/>
<div xml:id="echoid-div79" type="page_commentary" level="2" n="79">
<p>
<s xml:id="echoid-s457" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s457" xml:space="preserve">
The method of solution here is essentially the same as that on Add MS 6783, f. 145.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head76" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (6.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s459" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f140v" o="140v" n="280"/>
<pb file="add_6783_f141" o="141" n="281"/>
<div xml:id="echoid-div80" type="page_commentary" level="2" n="80">
<p>
<s xml:id="echoid-s460" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s460" xml:space="preserve">
The method of solution here is essentially the same as that on Add MS 6783, f. 145.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head77" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (5.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s462" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f141v" o="141v" n="282"/>
<pb file="add_6783_f142" o="142" n="283"/>
<div xml:id="echoid-div81" type="page_commentary" level="2" n="81">
<p>
<s xml:id="echoid-s463" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s463" xml:space="preserve">
This page gives several numerical examples of the type of equation treated in Add MS 6783, f. 143. <lb/>
Sheet d.11 (Add MS 6783, 173) shows the canonical form for this type of equation,
generated from the multiplication of linear factors. It is listed again on sheet d.14 (Add MS 6783, f. 170). <lb/>
Sheet c.14 (Add MS 6782, f. 404) again shows the canonical form for equations of this type. <lb/>
Sheet d.20 shows how to eliminate fractions from the coefficients by appropriate multiplications.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head78" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (4.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s465" xml:space="preserve">
Exempla antecedentis differentiæ in numeris
<lb/>[<emph style="it">tr: 
Examples of the preceding case in numbers
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s466" xml:space="preserve">
Sit æquatio <lb/>
adventitia
<lb/>[<emph style="it">tr: 
Let the equation in question be
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s467" xml:space="preserve">
(Vide. d. 11. <lb/>
vel: d.14. <lb/>
vel: c.14)
<lb/>[<emph style="it">tr: 
(See d.11 or d.14 or c.14)
</emph>]<lb/>
[<emph style="it">Note: 
Sheets d.11 and d.14 are Add MS 6783, f. 173 and f. 170; sheet c.14 is Add MS 6782, f. 404.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s468" xml:space="preserve">
multiplica per
<lb/>[<emph style="it">tr: 
multiply by
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s469" xml:space="preserve">
erit:
it will be:
</s>
<lb/>
<s xml:id="echoid-s470" xml:space="preserve">
multiplica æquationem per, 84 (ut d.20)
<lb/>[<emph style="it">tr: 
multiply the equation by 84 (as in d.20)
</emph>]<lb/>
[<emph style="it">Note: 
Sheet d.20 is Add MS 6783, f. 164.
 </emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s471" xml:space="preserve">
= coniugatæ: <lb/>
quod nota.
<lb/>[<emph style="it">tr: 
these are conjugates, to be noted.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s472" xml:space="preserve">
multiplica æquationem per, 28 (ut d.20)
<lb/>[<emph style="it">tr: 
multiply the equation by 28 (as in d.20)
</emph>]<lb/>
[<emph style="it">Note: 
Sheet d.20 is Add MS 6783, f. 164.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6783_f142v" o="142v" n="284"/>
<pb file="add_6783_f143" o="143" n="285"/>
<div xml:id="echoid-div82" type="page_commentary" level="2" n="82">
<p>
<s xml:id="echoid-s473" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s473" xml:space="preserve">
The method of solution here is the same as that on Add MS 6783, f. 145,
but here Harriot expects to find two real roots.
Numerical examples follow on Add MS 6783, f. 142.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head79" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (3.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s475" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s476" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> duplex
<lb/>[<emph style="it">tr: 
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f143v" o="143v" n="286"/>
<pb file="add_6783_f144" o="144" n="287"/>
<div xml:id="echoid-div83" type="page_commentary" level="2" n="83">
<p>
<s xml:id="echoid-s477" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s477" xml:space="preserve">
The method of solution here is the same as that on Add MS 6783, f. 145.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head80" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (2.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s479" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f144v" o="144v" n="288"/>
<pb file="add_6783_f145" o="145" n="289"/>
<div xml:id="echoid-div84" type="page_commentary" level="2" n="84">
<p>
<s xml:id="echoid-s480" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s480" xml:space="preserve">
This is the first page of a 7-page section on removing the cube term from a quartic equation;
this is the 'reduction' that appears in the headings throughout Section f.
Harriot's friend and colleague Nathaniel Torporley later denoted this subsection by the letter β.
It is not self-contained, since the pagination (here 1 to 7) later continues into section γ (8 to 19),
but it is convenient to treat these first seven pages together.
There is also another version of it, for which Torporley used the letter α,
in Add MS 6783, f. 113 to f. 118 and f. 130. <lb/>
This first page shows the removal of the cube term from the quartic equations <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mo>+</mo><mn>4</mn><mi>r</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>,
using the substitution <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>e</mi><mo>-</mo><mi>r</mi></mstyle></math>. <lb/>
For another version of this page, see Add MS 6783, f. 161.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head81" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (1.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s482" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f145v" o="145v" n="290"/>
<pb file="add_6783_f146" o="146" n="291"/>
<div xml:id="echoid-div85" type="page_commentary" level="2" n="85">
<p>
<s xml:id="echoid-s483" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s483" xml:space="preserve">
This is the first page of a 4-page subsection of Section f,
in which Harriot continues to remove the cube term from a general quartic.
Harriot's colleague Torporley later denoted this subsection by the letter ε (see Stedall 2003, 284–287). <lb/>
As in subsection ϛ (Add MS 6783, f. 199 to f. 202) Harriot is able to treat four equations at a time.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head82" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s485" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f146v" o="146v" n="292"/>
<pb file="add_6783_f147" o="147" n="293"/>
<div xml:id="echoid-div86" type="page_commentary" level="2" n="86">
<p>
<s xml:id="echoid-s486" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s486" xml:space="preserve">
A continuation of Add MS 6783, f. 146.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head83" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s488" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f147v" o="147v" n="294"/>
<pb file="add_6783_f148" o="148" n="295"/>
<div xml:id="echoid-div87" type="page_commentary" level="2" n="87">
<p>
<s xml:id="echoid-s489" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s489" xml:space="preserve">
A continuation of Add MS 6783, f. 146 and f. 147.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head84" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s491" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f148v" o="148v" n="296"/>
<pb file="add_6783_f149" o="149" n="297"/>
<div xml:id="echoid-div88" type="page_commentary" level="2" n="88">
<p>
<s xml:id="echoid-s492" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s492" xml:space="preserve">
A continuation of Add MS 6783, f. 146, f. 147, and f. 148.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head85" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s494" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f149v" o="149v" n="298"/>
<pb file="add_6783_f150" o="150" n="299"/>
<div xml:id="echoid-div89" type="page_commentary" level="2" n="89">
<p>
<s xml:id="echoid-s495" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s495" xml:space="preserve">
This and the subsequent sheets, paginated by Harriot as f.8 to f.18,
continue the work begun in subsection α (beginning on Add MS 6783, f. 113)
and subsection β (beginning on Add MS 6783, f. 145), both of which consist of sheets paginated f.1 to f.7.
Harriot's colleague Torporley later denoted this section by the letter γ (see Stedall 2003, 259–275). <lb/>
Here Harriot removes the cube term from a quartic equation beginning with <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>4</mn><mi>r</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>,
using the substitution <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>r</mi><mo>=</mo><mi>e</mi></mstyle></math>. Numerical examples follow in Add MS 6783, f. 151 to f. 156. <lb/>
There are many other sheets in the manuscripts with the heading 'f.8',
suggesting that Harriot worked over this process many times.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head86" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (8.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s497" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f150v" o="150v" n="300"/>
<pb file="add_6783_f151" o="151" n="301"/>
<div xml:id="echoid-div90" type="page_commentary" level="2" n="90">
<p>
<s xml:id="echoid-s498" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s498" xml:space="preserve">
Numerical examples of the method shown in Add MS 6783, f. 150.
Harriot solves both equations for all four roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head87" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (9.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s500" xml:space="preserve">
Exempla antecedentis differentiæ in numeris.
<lb/>[<emph style="it">tr: 
Examples of the preceding cases in numbers
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f151v" o="151v" n="302"/>
<pb file="add_6783_f152" o="152" n="303"/>
<div xml:id="echoid-div91" type="page_commentary" level="2" n="91">
<p>
<s xml:id="echoid-s501" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s501" xml:space="preserve">
Further numerical examples of the method shown in Add MS 6783, f. 150.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head88" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (10.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s503" xml:space="preserve">
Exempla antecedentis differentiæ in numeris.
<lb/>[<emph style="it">tr: 
Examples of the preceding cases in numbers
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f152v" o="152v" n="304"/>
<pb file="add_6783_f153" o="153" n="305"/>
<div xml:id="echoid-div92" type="page_commentary" level="2" n="92">
<p>
<s xml:id="echoid-s504" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s504" xml:space="preserve">
Further numerical examples of the method shown in Add MS 6783, f. 150.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head89" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (11.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s506" xml:space="preserve">
Exempla antecedentis differentiæ in numeris.
<lb/>[<emph style="it">tr: 
Examples of the preceding cases in numbers
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f153v" o="153v" n="306"/>
<pb file="add_6783_f154" o="154" n="307"/>
<div xml:id="echoid-div93" type="page_commentary" level="2" n="93">
<p>
<s xml:id="echoid-s507" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s507" xml:space="preserve">
Further numerical examples of the method shown in Add MS 6783, f. 150.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head90" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (12.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s509" xml:space="preserve">
Exempla antecedentis differentiæ in numeris.
<lb/>[<emph style="it">tr: 
Examples of the preceding cases in numbers
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f154v" o="154v" n="308"/>
<pb file="add_6783_f155" o="155" n="309"/>
<div xml:id="echoid-div94" type="page_commentary" level="2" n="94">
<p>
<s xml:id="echoid-s510" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s510" xml:space="preserve">
Further numerical examples of the method shown in Add MS 6783, f. 150.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head91" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (13.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s512" xml:space="preserve">
Exempla antecedentis differentiæ in numeris.
<lb/>[<emph style="it">tr: 
Examples of the preceding cases in numbers
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f155v" o="155v" n="310"/>
<pb file="add_6783_f156" o="156" n="311"/>
<div xml:id="echoid-div95" type="page_commentary" level="2" n="95">
<p>
<s xml:id="echoid-s513" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s513" xml:space="preserve">
On this page Harriot continues the explorations begun in Add MS 6783, f. 178,
of quadratic factors of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>f</mi><mo>±</mo><mi>a</mi><mi>a</mi></mstyle></math>, but now in quartic equations.
He is explicit here about the possibility of negative or imaginary roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head92" xml:space="preserve" xml:lang="lat">
d.13.2<emph style="super">o</emph>.) De generatione æquationum canonicarum.
<lb/>[<emph style="it">tr: 
On the generation of canonical equations
</emph>]<lb/>
</head>
<pb file="add_6783_f156v" o="156v" n="312"/>
<pb file="add_6783_f157" o="157" n="313"/>
<div xml:id="echoid-div96" type="page_commentary" level="2" n="96">
<p>
<s xml:id="echoid-s515" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s515" xml:space="preserve">
On this page, for the first time in this section,
Harriot solves a quartic equation not only for two real roots but also for two imaginary roots.
He states that these are derived from the rules given on sheet e.13 (Add MS 6783, f.111).
There, and on e.14 (Add MS 6783, f. 112) he showed how to derive two roots of an equation
from a single known root of its conjugate. More specifically,
given the single positive root <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math> of an equation of the form <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>f</mi><mi>a</mi><mo>=</mo><mi>g</mi><mi>g</mi><mi>h</mi></mstyle></math>,
Harriot derived a quadratic equation for the two roots of the conjugate equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>f</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>g</mi><mi>g</mi><mi>h</mi></mstyle></math>.
That quadratic equation was <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>e</mi><mo>-</mo><mi>x</mi><mi>e</mi><mo>=</mo><mi>d</mi><mi>f</mi><mo>-</mo><mi>x</mi><mi>x</mi></mstyle></math>, with roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mfrac><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>±</mo><msqrt><mrow><mi>d</mi><mi>f</mi><mo>-</mo><mfrac><mrow><mn>3</mn><mi>x</mi><mi>x</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msqrt></mstyle></math>. <lb/>
On this page Harriot wants to solve the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>1</mn><mi>e</mi><mo>-</mo><mi>e</mi><mi>e</mi><mi>e</mi></mstyle></math>.
The conjugate equation is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>e</mi><mi>e</mi><mo>-</mo><mn>1</mn><mn>1</mn><mi>e</mi><mo>=</mo><mn>2</mn><mn>0</mn></mstyle></math>, which has the single positive root <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mn>4</mn></mstyle></math>
(as can be seen by inspection).
Harriot therefore now has <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mo>=</mo><mn>4</mn></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>f</mi><mo>=</mo><mn>1</mn><mn>1</mn></mstyle></math>.
Thus his quadratic equation is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>e</mi><mo>-</mo><mn>4</mn><mi>e</mi><mo>+</mo><mn>5</mn><mo>=</mo><mn>0</mn></mstyle></math>, with roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mn>2</mn><mo>±</mo><msqrt><mrow><mo>-</mo><mn>1</mn></mrow></msqrt></mstyle></math>.
Previously he might have rejected these roots as 'impossible' (see sheet e.16, Add MS 6783, f. 197,
but on this page he accepts them, and follows the working to its conclusion,
giving him all four roots of the original equation. <lb/>
He notes that roots of the kind he has found here are given by the canonical form
<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><mi>a</mi><mo>+</mo><mi>d</mi><mi>f</mi><mo maxsize="1">)</mo></mstyle></math> shown in sheet d.13.2 (Ad MS 6783, f. 156),
a sheet that has been inserted immediately before the present page.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head93" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (14.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s517" xml:space="preserve">
Exempla antecedentis differentiæ in numeris.
<lb/>[<emph style="it">tr: 
Examples of the preceding cases in numbers
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s518" xml:space="preserve">
Hæc æquatio non habet alius radices hypostaticus præter 2, et 6. <lb/>
quoniam æquales sunt coefficienti longitudini.
<lb/>[<emph style="it">tr: 
This equation has no other roots pertaining to it besides 2 and 6,
because these are equal to the longitudinal coefficient.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s519" xml:space="preserve">
Si essent plures, summa omnium esset maior.
<lb/>[<emph style="it">tr: 
If there were more, the sum of all of them would be greater.
</emph>]<lb/>
</s>
<s xml:id="echoid-s520" xml:space="preserve">
Quod est contra <lb/>
canonem quatuor radicum.
<lb/>[<emph style="it">tr: 
Which is against the canon for four roots.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s521" xml:space="preserve">
Attamen duas alias habet noeticas (saluo canone) quas <lb/>
ex reductione sequenti, adposui; causa exempli et ad <lb/>
ulterius contemplandum.
<lb/>[<emph style="it">tr: 
Nevertheless, it has two other imaginary roots (preserving the canon),
which I have set out in the following reduction; as an example and to be considered further.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s522" xml:space="preserve">
Reductio.
<lb/>[<emph style="it">tr: 
Reduction
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s523" xml:space="preserve">
radices istæ <lb/>
noeticæ habentur <lb/>
per (e.13.)
<lb/>[<emph style="it">tr: 
these imaginary roots are to be had by sheet e.13
</emph>]<lb/>
[<emph style="it">Note: 
Sheet e.13 is Add MS 6783, f.111.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s524" xml:space="preserve">
noeticæ <lb/>
radices
<lb/>[<emph style="it">tr: 
imaginary roots
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s525" xml:space="preserve">
Æquationes quadrato-quadraticæ quæ habent duas radices <lb/>
æquales coefficienti longitudini generantur, (d.13.2<emph style="super">o</emph>.)
<lb/>[<emph style="it">tr: 
Quartic equations which have two roots equal to the longitudinal coefficient may be generated, as in sheet d.13.2.
</emph>]<lb/>
[<emph style="it">Note: 
Sheet d.13.2 is Add MS 6783, f. 156. It has been placed immediately before the present page.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6783_f157v" o="157v" n="314"/>
<pb file="add_6783_f158" o="158" n="315"/>
<div xml:id="echoid-div97" type="page_commentary" level="2" n="97">
<p>
<s xml:id="echoid-s526" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s526" xml:space="preserve">
This is the only place, throughout Harriot's treatise on equations,
that there is any hint of mixing equations of different dimensions.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head94" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (15.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s528" xml:space="preserve">
Appendicula.
<lb/>[<emph style="it">tr: 
A short appendix
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s529" xml:space="preserve">
De inveniendis quibusdam æquationibus conditionalis in numeris, <lb/>
aliter suam in antecedentibus.
<lb/>[<emph style="it">tr: 
On finding equations under certain conditions in numbers, other than the preceding ones.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s530" xml:space="preserve">
Lemma. 1. Si duæ vel plures æquationes eiusdem vel diversuum <lb/>
specierum habent communes radices: summa, vel differetia illorum <lb/>
habebit easdem radices.
<lb/>[<emph style="it">tr: 
Lemma 1. If two or more equations of the same or different types have roots in common,
their sum or difference will have the same roots.
</emph>]<lb/>
</s>
<s xml:id="echoid-s531" xml:space="preserve">
Huisdmodo æquationes appello cognatas.
<lb/>[<emph style="it">tr: 
Equations of this kind I call cognates.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s532" xml:space="preserve">
Lemma. 2. Cuiuslibet æquationis termini, si numero quovis <lb/>
multiplicentur: facta æquatio habebit eandem radicem <lb/>
vel radices ut prima.
<lb/>[<emph style="it">tr: 
Lemma 2. If the terms of any equation are multiplied by any number,
the resulting equation will have the same root or roots as the first.
</emph>]<lb/>
</s>
<s xml:id="echoid-s533" xml:space="preserve">
hoc est, sunt illi cognata.
<lb/>[<emph style="it">tr: 
That is, they are cognates.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s534" xml:space="preserve">
Exempla utrinque lemmatis in numeris.
<lb/>[<emph style="it">tr: 
Examples of both lemmas in numbers.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f158v" o="158v" n="316"/>
<pb file="add_6783_f159" o="159" n="317"/>
<div xml:id="echoid-div98" type="page_commentary" level="2" n="98">
<p>
<s xml:id="echoid-s535" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s535" xml:space="preserve">
For another page with similar material see Add MS 6783, f. 57v.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head95" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (14.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s537" xml:space="preserve">
Appendicula.
<lb/>[<emph style="it">tr: 
A short appendix
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s538" xml:space="preserve">
Problema
<lb/>[<emph style="it">tr: 
Problem
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s539" xml:space="preserve">
Sit æquationis forma data:
<lb/>[<emph style="it">tr: 
Let the form of the equation be given:
</emph>]<lb/>
</s>
<s xml:id="echoid-s540" xml:space="preserve">
quatuor radicum potus est.
<lb/>[<emph style="it">tr: 
it may have four roots
</emph>]<lb/>
</s>
<s xml:id="echoid-s541" xml:space="preserve">
Oportet invenire æquationum <lb/>
sub eadem forma in numeris ut habent solummodo radices duas.
<lb/>[<emph style="it">tr: 
It is necessary to find equations in the same form in numbers that have only two roots.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s542" xml:space="preserve">
Quoniam potestas est negativa, non feret unicam vel tres singulas.
<lb/>[<emph style="it">tr: 
Because the leading power is negative, it cannot have one or three individual roots.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s543" xml:space="preserve">
æquatio <lb/>
quæsita
<lb/>[<emph style="it">tr: 
the equations sought
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s544" xml:space="preserve">
Problema soluitur brevius
<lb/>[<emph style="it">tr: 
The problem solved more briefly.
</emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s545" xml:space="preserve">
æquatio quæsita conditionis, nam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>r</mi><mi>r</mi><mo>&lt;</mo><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
the conditional equation sought, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>r</mi><mi>r</mi><mo>&lt;</mo><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f159v" o="159v" n="318"/>
<pb file="add_6783_f160" o="160" n="319"/>
<div xml:id="echoid-div99" type="page_commentary" level="2" n="99">
<p>
<s xml:id="echoid-s546" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s546" xml:space="preserve">
The same method of reduction as on Add MS 6783, f. 150.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head96" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (17.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s548" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f160v" o="160v" n="320"/>
<pb file="add_6783_f161" o="161" n="321"/>
<div xml:id="echoid-div100" type="page_commentary" level="2" n="100">
<p>
<s xml:id="echoid-s549" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s549" xml:space="preserve">
Another version of Add MS 6783, f. 145.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head97" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (1.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s551" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f161v" o="161v" n="322"/>
<pb file="add_6783_f162" o="162" n="323"/>
<div xml:id="echoid-div101" type="page_commentary" level="2" n="101">
<p>
<s xml:id="echoid-s552" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s552" xml:space="preserve">
Numerical examples for the method shown on Add MS 6783, f. 161.
In both cases the equation are solved for all four roots, two real and two imaginary.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head98" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem. (18.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s554" xml:space="preserve">
Exempla antecedentis differentiæ in numeris.
<lb/>[<emph style="it">tr: 
Examples of the preceding cases in numbers
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s555" xml:space="preserve">
In istis exemplis, summa duarum radicum æqualis est <lb/>
coefficienti longitudini. <lb/>
ita etiam summa omnium quatuor
<lb/>[<emph style="it">tr: 
In these examples, the sum of the two roots is equal to the longitudinal coefficient; <lb/>
and thus also the sum of all four
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s556" xml:space="preserve">
Huiusmodi æquationes generantur, (d.13.2<emph style="super">o</emph>.)
<lb/>[<emph style="it">tr: 
Equations of this kind may be generated, as in sheet d.13.2.
</emph>]<lb/>
[<emph style="it">Note: 
Sheet d.13.2 is Add MS 6783, f. 156.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6783_f162v" o="162v" n="324"/>
<pb file="add_6783_f163" o="163" n="325"/>
<div xml:id="echoid-div102" type="page_commentary" level="2" n="102">
<p>
<s xml:id="echoid-s557" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s557" xml:space="preserve">
This page repeats the calculations already shown on the previous page, Add MS 6783, f. 164.
Here, however, it is particularly used to explain how, by appropriate multiplication by powers of 10,
a root may be obtained to increasing degrees of accuracy.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head99" xml:space="preserve" xml:lang="lat">
d.21.) De generatione æquationum canonicarum.
<lb/>[<emph style="it">tr: 
On the generation of canonical equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s559" xml:space="preserve">
2. Appendicula. De multiplicatione radicum.
<lb/>[<emph style="it">tr: 
Appendix 2. On the multiplication of roots.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s560" xml:space="preserve">
In data æquatio
<lb/>[<emph style="it">tr: 
In the given equation
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s561" xml:space="preserve">
Sit radix multiplicanda <lb/>
per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
Let the root be multiplied by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s562" xml:space="preserve">
Sunt igitur proportionales.
<lb/>[<emph style="it">tr: 
These are therefore proportionals.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s563" xml:space="preserve">
Dico quod in hac æquatione: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>d</mi><mi>a</mi></mstyle></math>, superioris.
<lb/>[<emph style="it">tr: 
I say that in this equation, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>a</mi></mstyle></math> above.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s564" xml:space="preserve">
Et ita est. Est igitur.
<lb/>[<emph style="it">tr: 
And so it is. Therefore it is so.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s565" xml:space="preserve">
Exempla in numeris.
<lb/>[<emph style="it">tr: 
Examples in numbers.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s566" xml:space="preserve">
In data æquati-<lb/>
one sit radix <lb/>
<emph style="super">quælibet</emph> duplicanda.
<lb/>[<emph style="it">tr: 
In the given equation let any root be doubled.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s567" xml:space="preserve">
Alia exempla in numeris.
<lb/>[<emph style="it">tr: 
Another example in numbers.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s568" xml:space="preserve">
Decuplatio <lb/>
radicis.
<lb/>[<emph style="it">tr: 
Multiplication of the root by 10.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s569" xml:space="preserve">
Et sic de alijs omnibus æquationibus.
<lb/>[<emph style="it">tr: 
And so on in all other equations.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s570" xml:space="preserve">
Consectarium.
<lb/>[<emph style="it">tr: 
Consequence
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s571" xml:space="preserve">
Ex decuplatione et cen-<lb/>
tuplatione &amp;c radicis <lb/>
apparet quomodo fractiones <lb/>
denaria, centenaria &amp;c. <lb/>
habeantur in resolutionibus <lb/>
videlicet addendo cyphras <lb/>
et puncta legitime, ut <lb/>
potestates et coefficientes <lb/>
postulant.
<lb/>[<emph style="it">tr: 
From multiplication by 10, 100, etc. it is clear how tenth, hundredth, etc. fractions in the root
may be had in the solution, evidently by adding zeros and correct points, as the powers and coefficients require.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f163v" o="163v" n="326"/>
<pb file="add_6783_f164" o="164" n="327"/>
<div xml:id="echoid-div103" type="page_commentary" level="2" n="103">
<p>
<s xml:id="echoid-s572" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s572" xml:space="preserve">
Here Harriot shows how to eliminate fractions in the coefficients of an equation by an appropriate substitution.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head100" xml:space="preserve" xml:lang="lat">
d.20.) De generatione æquationum canonicarum.
<lb/>[<emph style="it">tr: 
On the generation of canonical equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s574" xml:space="preserve">
2. Appendicula. De multiplicatione radicum. <lb/>[<emph style="it">tr: 
Appendix 2. On the multiplication of roots
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s575" xml:space="preserve">
Sit æquatio adventitia.
<lb/>[<emph style="it">tr: 
Let the equation in question be
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s576" xml:space="preserve">
Tum per parabolismum:
<lb/>[<emph style="it">tr: 
Then by removal of excess letters:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s577" xml:space="preserve">
Si in numeris, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> non <lb/>
dividat numerus supra <lb/>
positis; oportet multiplicare <lb/>
<emph style="st">radicem</emph> illos per continue <lb/>
proportionales, ita:
<lb/>[<emph style="it">tr: 
If in numbers <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> does not divide the number supposed above, we must multiply them by continued proportionals.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s578" xml:space="preserve">
Dico quod in hac æquatione: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>d</mi><mi>a</mi></mstyle></math>, superioris.
<lb/>[<emph style="it">tr: 
I say that in this equation, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>a</mi></mstyle></math> above.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s579" xml:space="preserve">
Et ita est. Est igitur.
<lb/>[<emph style="it">tr: 
And so it is. Therefore it is so.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s580" xml:space="preserve">
Exempla in numeris.
<lb/>[<emph style="it">tr: 
Examples in numbers.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s581" xml:space="preserve">
Tum mult: <lb/>
per:
<lb/>[<emph style="it">tr: 
Then multiplying by:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s582" xml:space="preserve">
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>6</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>=</mo><mn>2</mn><mo>=</mo><mi>a</mi></mstyle></math>, prioris æqationes.
<lb/>[<emph style="it">tr: 
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>6</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>=</mo><mn>2</mn><mo>=</mo><mi>a</mi></mstyle></math>, in the first equations.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f164v" o="164v" n="328"/>
<pb file="add_6783_f165" o="165" n="329"/>
<div xml:id="echoid-div104" type="page_commentary" level="2" n="104">
<p>
<s xml:id="echoid-s583" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s583" xml:space="preserve">
The calculations on this page are similar to those on Add MS 6783, f. 167 and f. 166,
here for whole numbers of the forms <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>b</mi><mi>b</mi><mo>+</mo><mi>c</mi><mi>c</mi></mrow><mrow><mi>b</mi><mo>+</mo><mi>c</mi></mrow></mfrac></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>c</mi></mrow><mrow><mi>b</mi><mo>+</mo><mi>c</mi></mrow></mfrac></mstyle></math>.
In the latter case, Harriot notes that
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>c</mi></mrow><mrow><mi>b</mi><mo>+</mo><mi>c</mi></mrow></mfrac><mo>=</mo><mi>b</mi><mi>b</mi><mo>-</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi></mstyle></math> and that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mi>c</mi><mi>c</mi><mi>c</mi></mrow><mrow><mi>b</mi><mo>-</mo><mi>c</mi></mrow></mfrac><mo>=</mo><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi></mstyle></math>,
as he also noted on Add MS 6784, f. 323.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head101" xml:space="preserve" xml:lang="lat">
d.19.) De generatione æquationum canonicarum.
<lb/>[<emph style="it">tr: 
On the generation of canonical equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s585" xml:space="preserve">
1. Appendicula.
<lb/>[<emph style="it">tr: 
Appendix 1
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s586" xml:space="preserve">
Invenire numerum:
<lb/>[<emph style="it">tr: 
To find numbers of the form:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s587" xml:space="preserve">
Quæsitum in specie,
<lb/>[<emph style="it">tr: 
What was sought, in letters.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s588" xml:space="preserve">
Invenire numerum:
<lb/>[<emph style="it">tr: 
To find numbers of the form:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s589" xml:space="preserve">
Quæsitæ in specie, <lb/>
ut supra, erit:
<lb/>[<emph style="it">tr: 
What was sought, in letters, as above, will be:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s590" xml:space="preserve">
Etsi species antecedens sit vera:
<lb/>[<emph style="it">tr: 
Although the preceding rule was true:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s591" xml:space="preserve">
Atque igitur est sua natura <lb/>
<emph style="st">species</emph> numero in specie.
<lb/>[<emph style="it">tr: 
And therefore the characteristic of the number in letters is.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s592" xml:space="preserve">
Atque hoc, est insigne poristicum <lb/>
cuius usus erit aliquando in <lb/>
sequentibus. et sæpe alias.
<lb/>[<emph style="it">tr: 
And here, is a notable fact which wll be used sometimes in what follows, and often elsewhere.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f165v" o="165v" n="330"/>
<pb file="add_6783_f166" o="166" n="331"/>
<div xml:id="echoid-div105" type="page_commentary" level="2" n="105">
<p>
<s xml:id="echoid-s593" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s593" xml:space="preserve">
The continuation of Add MS 6783, f. 167.
Here Harriot seeks whole numbers of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>=</mo><mfrac><mrow><mi>b</mi><mi>c</mi><mi>d</mi></mrow><mrow><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>d</mi><mo>+</mo><mi>c</mi><mi>d</mi></mrow></mfrac></mstyle></math>,
as required for sheet d. 8, that is, Add MS 6783, f. 176, and also of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>b</mi><mi>c</mi><mi>d</mi></mrow><mrow><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfrac></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head102" xml:space="preserve" xml:lang="lat">
d.18.) De generatione æquationum canonicarum.
<lb/>[<emph style="it">tr: 
On the generation of canonical equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s595" xml:space="preserve">
1. Appendicula.
<lb/>[<emph style="it">tr: 
Appendix 1
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s596" xml:space="preserve">
d.8) Invenire numerum:
<lb/>[<emph style="it">tr: 
To find numbers of the form:
</emph>]<lb/>
[<emph style="it">Note: 
Page d.8 is Add MS 6783, f. 176.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s597" xml:space="preserve">
Quæsitum in specie,
<lb/>[<emph style="it">tr: 
What was sought, in letters.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s598" xml:space="preserve">
Invenire numerum:
<lb/>[<emph style="it">tr: 
To find numbers of the form:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s599" xml:space="preserve">
Quæsitum in specie,
<lb/>[<emph style="it">tr: 
What was sought, in letters.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f166v" o="166v" n="332"/>
<pb file="add_6783_f167" o="167" n="333"/>
<div xml:id="echoid-div106" type="page_commentary" level="2" n="106">
<p>
<s xml:id="echoid-s600" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s600" xml:space="preserve">
The canonical equation derived on sheet d.5 (Add MS 6783, f. 179) has a root of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mfrac><mrow><mi>b</mi><mi>c</mi></mrow><mrow><mi>b</mi><mo>+</mo><mi>c</mi></mrow></mfrac></mstyle></math>.
In this page, Harriot seeks to discover whole numbers of this form.
It is not clear why he proceeds by means of ratios rather than simply trying out values 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>
as he does at the end. <lb/>
Similar calculations follow for the root <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>=</mo><mfrac><mrow><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>d</mi><mo>+</mo><mi>c</mi><mi>d</mi></mrow><mrow><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi></mrow></mfrac></mstyle></math> on sheet c.8,
that is, Add MS 6783, f. 176. This work is continued on the next page, Add MS 6783, f. 166.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head103" xml:space="preserve" xml:lang="lat">
d.17.) De generatione æquationum canonicarum.
<lb/>[<emph style="it">tr: 
On the generation of canonical equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s602" xml:space="preserve">
1. Appendicula. De inveniendis numeros <lb/>
quibusdam conditionalibus.
<lb/>[<emph style="it">tr: 
Appendix 1. On finding numbers satisfying certain conditions
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s603" xml:space="preserve">
d.5) Invenire numerum:
<lb/>[<emph style="it">tr: 
To find numbers of the form:
</emph>]<lb/>
[<emph style="it">Note: 
Page d.5 is Add MS 6783, f. 179.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s604" xml:space="preserve">
ergo quæsitum in specie,
<lb/>[<emph style="it">tr: 
Therefore, what was sought, in letters.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s605" xml:space="preserve">
d.8) Invenire numerum:
<lb/>[<emph style="it">tr: 
To find numbers of the form:
</emph>]<lb/>
[<emph style="it">Note: 
Page d.8 is Add MS 6783, f. 176.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s606" xml:space="preserve">
Quæsitum in specie,
<lb/>[<emph style="it">tr: 
What was sought, in letters.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f167v" o="167v" n="334"/>
<pb file="add_6783_f168" o="168" n="335"/>
<div xml:id="echoid-div107" type="page_commentary" level="2" n="107">
<p>
<s xml:id="echoid-s607" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s607" xml:space="preserve">
Another collection of canonical equations, as on Add MS 6783, f. 170 and f. 169,
but now for equations with as many positive roots as the degree of the equation.
The last of these, an equation of fith degree has not been derived in the pages of Section d,
but can be written down by analogy; see also Add MS 6784, f. 427. <lb/>
Viète, at the end of his <emph style="it">De recogntione et emendatione æquationum, tractatus duo</emph> (1615),
offered several collections of equations and their solutions.
In general Harriot's collections are different from Viète's.
This page, however, contains the same equations as Viète's fourth and final collection, the 'Collectio quarta'.
I would argue that Viète derived his equations by his method of <emph style="it">syncrisis</emph>
(see Stedall 2011, 24–27), which is quite different from Harriot's multiplication of factors.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head104" xml:space="preserve" xml:lang="lat">
d.16.) De generatione æquationum canonicarum.
<lb/>[<emph style="it">tr: 
On the generation of canonical equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s609" xml:space="preserve">
Alia collectio et series canonicarum.
<lb/>[<emph style="it">tr: 
Another collection and series of canonical equations.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f168v" o="168v" n="336"/>
<pb file="add_6783_f169" o="169" n="337"/>
<div xml:id="echoid-div108" type="page_commentary" level="2" n="108">
<p>
<s xml:id="echoid-s610" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s610" xml:space="preserve">
A collection of canonical equations, similar to that on Add MS 6783, f. 170,
but now for equations with three positive roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head105" xml:space="preserve" xml:lang="lat">
d.15.) De generatione æquationum canonicarum.
<lb/>[<emph style="it">tr: 
On the generation of canonical equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s612" xml:space="preserve">
Alia collectio et series canonicarum.
<lb/>[<emph style="it">tr: 
Another collection and series of canonical equations.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s613" xml:space="preserve">
Et sic de alijs ut opus est.
<lb/>[<emph style="it">tr: 
And so on for others, as needed.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f169v" o="169v" n="338"/>
<pb file="add_6783_f170" o="170" n="339"/>
<div xml:id="echoid-div109" type="page_commentary" level="2" n="109">
<p>
<s xml:id="echoid-s614" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s614" xml:space="preserve">
This page summarizes some of the canonical equations derived so far, but also extends them to higher degrees.
The first three, quadratic, cubic, and quartic, for example, were derived on sheets d.1, d.3, d.10 respectively,
but the fifth degree equation that follows is written down purely by analogy. <lb/>
All the equations on this page have two positive roots, and so are particularly relevant to Section c.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head106" xml:space="preserve" xml:lang="lat">
d.14.) De generatione æquationum canonicarum.
<lb/>[<emph style="it">tr: 
On the generation of canonical equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s616" xml:space="preserve">
Collectio aliquarum æquationum cum tali dispositione <lb/>
ut de facili apparent generatio aliarum, sublimiorum graduum.
<lb/>[<emph style="it">tr: 
A collection of other equations of similar layout, that there might more easily appear the generation of others,
of higher degree.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s617" xml:space="preserve">
Et sic de cæteris ut opus est.
<lb/>[<emph style="it">tr: 
And so on for others, as needed.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f170v" o="170v" n="340"/>
<pb file="add_6783_f171" o="171" n="341"/>
<div xml:id="echoid-div110" type="page_commentary" level="2" n="110">
<p>
<s xml:id="echoid-s618" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s618" xml:space="preserve">
Just as on Add MS 6783, f. 178 Harriot began to explore quadratic factors of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>±</mo><mi>b</mi><mi>c</mi></mstyle></math>,
here he explores the use of cubic factors of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>±</mo><mi>c</mi><mi>d</mi><mi>f</mi></mstyle></math>. Again these result in what Harriot calls
'reciprocal equations'.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head107" xml:space="preserve" xml:lang="lat">
d.13.) De generatione æquationum canonicarum.
<lb/>[<emph style="it">tr: 
On the generation of canonical equations
</emph>]<lb/>
De reciprocis æquationibus.
</head>
<p xml:lang="lat">
<s xml:id="echoid-s620" xml:space="preserve">
Radix hic nota est sive resolutione <lb/>
ut in cubicis æquationibus similis conditionis. (d.6.)
<lb/>[<emph style="it">tr: 
The root is known without solving as in cubic equations under similar conditions (d.6)
</emph>]<lb/>
[<emph style="it">Note: 
Page d.6 is Add MS 6783, f. 178.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6783_f171v" o="171v" n="342"/>
<pb file="add_6783_f172" o="172" n="343"/>
<div xml:id="echoid-div111" type="page_commentary" level="2" n="111">
<p>
<s xml:id="echoid-s621" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s621" xml:space="preserve">
On this page Harriot continues his exploration <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><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>f</mi><mo maxsize="1">)</mo><mo>=</mo><mn>0</mn></mstyle></math>
from Add MS 6783, f. 175, f. 174, and f. 173. <lb/>
On the two previous pages he demonstrated the conditions required for the terms in
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, or <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>a</mi><mi>a</mi></mstyle></math>, to vanish simultaneously.
Now he investigates the conditions for the third case, for both <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>a</mi><mi>a</mi><mi>a</mi></mstyle></math> to vanish,
and finds the resulting form of the coefficients in terms of the 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>.
He notes that his findings are backed up by the practical calculations given in sheet c. 14,
that is, Add MS 6782, f. 404
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head108" xml:space="preserve" xml:lang="lat">
d.12.) De generatione æquationum canonicarum
<lb/>[<emph style="it">tr: 
On the generation of canonical equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s623" xml:space="preserve">
3<emph style="super">o</emph> et ultimo. De ortu illius binomiæ quæ fit tollendo primum <lb/>
et tertium gradum, nempe (<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>a</mi><mi>a</mi></mstyle></math>.)
<lb/>[<emph style="it">tr: 
3, and last. On the generation of those binomials which arise from removing the first and third degree terms
(that is, <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>a</mi><mi>a</mi><mi>a</mi></mstyle></math>)
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s624" xml:space="preserve">
Ponendum.
<lb/>[<emph style="it">tr: 
Suppose.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s625" xml:space="preserve">
Æquatio quadratica <lb/>
duorum laterum
<lb/>[<emph style="it">tr: 
A quadratic equation with two roots.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s626" xml:space="preserve">
nimirum dato <lb/>
homogeneo <lb/>
quadraticæ æquationis.
<lb/>[<emph style="it">tr: 
Clearly the given homogene of the quadratic equation.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s627" xml:space="preserve">
Tum coefficientes secundi gradus <lb/>
nempe (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>) ita reducuntur.
<lb/>[<emph style="it">tr: 
Then the coefficients of the second degree term (that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>) are reduced thus.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s628" xml:space="preserve">
Datum homogeneum, ita:
<lb/>[<emph style="it">tr: 
The given homogene, thus:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s629" xml:space="preserve">
Ergo. Quæsita <lb/>
æquation binomia.
<lb/>[<emph style="it">tr: 
Therefore the sought binomial equation.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s630" xml:space="preserve">
Aliter, pro parte operationis <lb/>
antecedentis
<lb/>[<emph style="it">tr: 
Another way, for part of the preceding operation.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s631" xml:space="preserve">
Quadratica æquatio <lb/>
duorum laterum.
<lb/>[<emph style="it">tr: 
A quadratic equation with two sides.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s632" xml:space="preserve">
Quoniam:
<lb/>[<emph style="it">tr: 
Because:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s633" xml:space="preserve">
binomio cum suo <lb/>
residuo.
<lb/>[<emph style="it">tr: 
A binomial with itw residual.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s634" xml:space="preserve">
Dato homogeneo.
<lb/>[<emph style="it">tr: 
The given homogene.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s635" xml:space="preserve">
Tum coefficientes secundi gradus <lb/>
nempe (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>), ita reducuntur.
<lb/>[<emph style="it">tr: 
Then the coefficients of the seocnd degree term (that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>) are reduced thus.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s636" xml:space="preserve">
Datum homogeneum, ita:
<lb/>[<emph style="it">tr: 
The given homogene thus:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s637" xml:space="preserve">
ergo: Quæsita æquatio binomia: <lb/>
erit:
<lb/>[<emph style="it">tr: 
Therefore the sought binomial equation will be:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s638" xml:space="preserve">
Altera præcedens æquatio eadem est <lb/>
cum hac si reducatur.
<lb/>[<emph style="it">tr: 
Another preceding equation is the same as that so reduced.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s639" xml:space="preserve">
Utraque forma utilis suo loco et <lb/>
tempore.
<lb/>[<emph style="it">tr: 
Either form is useful in its own place and time.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f172v" o="172v" n="344"/>
<pb file="add_6783_f173" o="173" n="345"/>
<div xml:id="echoid-div112" type="page_commentary" level="2" n="112">
<p>
<s xml:id="echoid-s640" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s640" xml:space="preserve">
On this page Harriot continues his exploration <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><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>f</mi><mo maxsize="1">)</mo><mo>=</mo><mn>0</mn></mstyle></math>
from Add MS 6783, f. 175 and f. 174. <lb/>
On the previous page he demonstrated the conditions required for the terms in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> to vanish simultaneously;
now he investigates the conditions for both <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>a</mi><mi>a</mi></mstyle></math> to vanish,
and finds the resulting form of the coefficients in terms of the 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>.
He notes that his findings are backed up by the practical calculations given in sheet c. 14,
that is, Add MS 6782, f. 404
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head109" xml:space="preserve" xml:lang="lat">
d.11.) De generatione æquationum canonicarum
<lb/>[<emph style="it">tr: 
On the generation of canonical equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s642" xml:space="preserve">
2<emph style="super">o</emph>, De ortu illius binomiæ quæ fit tollendo primum <lb/>
et secundum gradum, nempe (<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>a</mi></mstyle></math>.)
<lb/>[<emph style="it">tr: 
2. On the generation of those binomials which arise from removing the first and second degree terms
(that is, <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>a</mi><mi>a</mi></mstyle></math>)
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s643" xml:space="preserve">
ponendum,
<lb/>[<emph style="it">tr: 
suppose:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s644" xml:space="preserve">
ergo. 1<emph style="super">o</emph>
<lb/>[<emph style="it">tr: 
therefore 1.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s645" xml:space="preserve">
æquatio quadratica <lb/>
duorum laterum
<lb/>[<emph style="it">tr: 
A quadratic equation with two roots.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s646" xml:space="preserve">
2<emph style="super">o</emph>.
<lb/>[<emph style="it">tr: 
2.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s647" xml:space="preserve">
Unde:
<lb/>[<emph style="it">tr: 
Whence:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s648" xml:space="preserve">
Tum, simili argumentatione superiori <lb/>
vel omnino eadem mutando <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>f</mi></mstyle></math>. <lb/>
concludetur <emph style="super">eadem</emph> æquatio quadratica duorum <lb/>
laterum.
<lb/>[<emph style="it">tr: 
Then, by a similar argument to that above, or everywhere changing <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>f</mi></mstyle></math>,
we end up with the same quadratic equation with two roots.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s649" xml:space="preserve">
eadem latera quam supra.
<lb/>[<emph style="it">tr: 
the same roots as above.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s650" xml:space="preserve">
Ergo si <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>, sunt diversi latera <lb/>
quod ponatur:
<lb/>[<emph style="it">tr: 
Therefore is <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> are different roots, as may be supposed:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s651" xml:space="preserve">
Erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>+</mo><mi>f</mi><mo>=</mo></mstyle></math> binomial + suo residuo
<lb/>[<emph style="it">tr: 
Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>+</mo><mi>f</mi></mstyle></math> is a binomial plus its residual.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s652" xml:space="preserve">
nimirum dato <lb/>
homogeneo.
<lb/>[<emph style="it">tr: 
clearly the given homogene
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s653" xml:space="preserve">
nam in quadraticis æquationibus ubi <lb/>
potestas negatur: facta e duobus <lb/>
lateribus æquatur dato homogeneo.
<lb/>[<emph style="it">tr: 
for in quadratic equations where the power is negative, the product of the two roots is equal to the given homogene.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s654" xml:space="preserve">
Tum coefficientes tertij gradus <lb/>
nempe (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>), ita reducuntur.
<lb/>[<emph style="it">tr: 
Then the coefficients of the third degree term (that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>), are reduced thus.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s655" xml:space="preserve">
Datum homogeneum, ita:
<lb/>[<emph style="it">tr: 
The given homogene, thus:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s656" xml:space="preserve">
Ergo, quæsita binomia æquatio, <lb/>
erit:
<lb/>[<emph style="it">tr: 
Therefore the sought binomial equation will be:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s657" xml:space="preserve">
Datum homogeneum, ita:
<lb/>[<emph style="it">tr: 
The given homogene thus:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s658" xml:space="preserve">
Hæc æquatio est vera et demonstra-<lb/>
tur (c.14.).
<lb/>[<emph style="it">tr: 
This equation is true and demonstrated in (c.14).
</emph>]<lb/>
[<emph style="it">Note: 
Page c.14 is Add MS 6782, f. 405.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s659" xml:space="preserve">
Ergo, ea quæ posita sunt ad eius <lb/>
investigationem; sunt etiam vera.
<lb/>[<emph style="it">tr: 
Therefore, those things that were supposed for this investigation are also true.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f173v" o="173v" n="346"/>
<pb file="add_6783_f174" o="174" n="347"/>
<div xml:id="echoid-div113" type="page_commentary" level="2" n="113">
<p>
<s xml:id="echoid-s660" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s660" xml:space="preserve">
On this page Harriot continues his exploration <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><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>f</mi><mo maxsize="1">)</mo><mo>=</mo><mn>0</mn></mstyle></math>
from Add MS 6783, f. 175. <lb/>
There he demonstrated the conditions required for the terms in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> to vanish;
here he works out the third case, in which <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> vanishes. <lb/>
He follows this, as on Add MS 6783, f. 204, by investigating the removal of two terms simultaneously,
those in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>,
and finds the resulting form of the coefficients in terms of the 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>. <lb/>
The work is continued on Add MS 6783, f. 173.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head110" xml:space="preserve" xml:lang="lat">
d.10.) De generatione æquationum canonicarum
<lb/>[<emph style="it">tr: 
On the generation of canonical equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s662" xml:space="preserve">
Si: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>d</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>f</mi><mo>=</mo><mi>b</mi><mi>d</mi><mi>f</mi><mo>+</mo><mi>c</mi><mi>d</mi><mi>f</mi></mstyle></math>. Tollitur gradus primus, nempe (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>). <lb/>
Et per reductionem, <lb/>
fit:
<lb/>[<emph style="it">tr: 
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>d</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>f</mi><mo>=</mo><mi>b</mi><mi>d</mi><mi>f</mi><mo>+</mo><mi>c</mi><mi>d</mi><mi>f</mi></mstyle></math>, the first degree term (that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>) may be removed,
whence by reduction there comes:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s663" xml:space="preserve">
De ortu binomiarum ex antecedenti primaria æquatione (d.9). <lb/>
et primo de illa quæ fit tollendo secundum et tertium gradum, <lb/>
nempe (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>.)
<lb/>[<emph style="it">tr: 
On the generation of binomials from the preceding primary equation (d.9), and first on that which arises
from removal of the second and third degree terms (that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>).
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s664" xml:space="preserve">
Ponendum:
 <lb/>[<emph style="it">tr: 
Suppose:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s665" xml:space="preserve">
seu radix
<lb/>[<emph style="it">tr: 
or the root
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s666" xml:space="preserve">
et quoniam
<lb/>[<emph style="it">tr: 
and because
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s667" xml:space="preserve">
et est altera radix æquationis.
<lb/>[<emph style="it">tr: 
and it is another root of the equation
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s668" xml:space="preserve">
Quoniam:
<lb/>[<emph style="it">tr: 
Because:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s669" xml:space="preserve">
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>f</mi><mo>=</mo><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mo>=</mo></mstyle></math> dato homogeneo <lb/>
quod oportet fieri,
<lb/>[<emph style="it">tr: 
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>f</mi><mo>=</mo><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi></mstyle></math> which is equal to the given homogene.
</emph>]<lb/>
[<emph style="it">Note: 
The 'datum homogeneum' is the given known term of the original equation.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s670" xml:space="preserve">
Quoniam:
<lb/>[<emph style="it">tr: 
Because:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s671" xml:space="preserve">
Tum coefficientes primi gradus ita redu-<lb/>
cuntur.
<lb/>[<emph style="it">tr: 
Then the coefficients of the first degree term may be reduced.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s672" xml:space="preserve">
Ergo, Quæsita binomia erit:
<lb/>[<emph style="it">tr: 
Therefore, the sought binomial is:
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f174v" o="174v" n="348"/>
<pb file="add_6783_f175" o="175" n="349"/>
<div xml:id="echoid-div114" type="page_commentary" level="2" n="114">
<p>
<s xml:id="echoid-s673" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s673" xml:space="preserve">
On this page Harriot explores 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><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>f</mi><mo maxsize="1">)</mo><mo>=</mo><mn>0</mn></mstyle></math>.
This is an equation of the kind that particularly interests him, with 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>. <lb/>
As elsewhere, he demonstrates the conditions required for the terms in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> to vanish.
The third case, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, follows on Add MS 6783, f. 174.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head111" xml:space="preserve" xml:lang="lat">
d.9.) De generatione æquationum canonicarum
<lb/>[<emph style="it">tr: 
On the generation of canonical equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s675" xml:space="preserve">
Ex hac æquatione <lb/>
oriuntur trino-<lb/>
mia sequentes, <lb/>
et præter tres binomia.
<lb/>[<emph style="it">tr: 
From this equation arise the following trinomials, and also three binomials.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s676" 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><mo>=</mo><mi>f</mi></mstyle></math>. Tollitur tertius gradus, nempe (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>). <lb/>
Et per reductionem, <lb/>
fit:
<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><mo>=</mo><mi>f</mi></mstyle></math>, the third degree term (that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>) may be removed,
whence by reduction there comes:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s677" xml:space="preserve">
<emph style="st">
Si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>&gt;</mo><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> <lb/>
fit antithesis <lb/>
vel signorum <lb/>
mutatio.
</emph>
<lb/>[<emph style="it">tr: 
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>&gt;</mo><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> it becomes the opposite with a change of sign.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s678" xml:space="preserve">
Si: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>+</mo><mi>d</mi><mi>f</mi><mo>=</mo><mi>b</mi><mi>d</mi><mo>+</mo><mi>c</mi><mi>d</mi><mo>+</mo><mi>b</mi><mi>f</mi><mo>+</mo><mi>c</mi><mi>f</mi></mstyle></math>. Tollitur secundus gradus, nempe (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>).
<lb/>[<emph style="it">tr: 
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>+</mo><mi>d</mi><mi>f</mi><mo>=</mo><mi>b</mi><mi>d</mi><mo>+</mo><mi>c</mi><mi>d</mi><mo>+</mo><mi>c</mi><mi>f</mi></mstyle></math>, the second degree term (that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>) may be removed.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s679" xml:space="preserve">
Tum primo. <lb/>
Unde.
<lb/>[<emph style="it">tr: 
Then the first is <lb/>
whence
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s680" xml:space="preserve">
Et per reductionem, <lb/>
fit:
<lb/>[<emph style="it">tr: 
And by reduction there comes:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s681" xml:space="preserve">
Tum secundo. <lb/>
Unde.
<lb/>[<emph style="it">tr: 
Then the second is <lb/>
whence
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s682" xml:space="preserve">
Et per reductionem, <lb/>
fit:
<lb/>[<emph style="it">tr: 
And by reduction there comes:
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f175v" o="175v" n="350"/>
<pb file="add_6783_f176" o="176" n="351"/>
<div xml:id="echoid-div115" type="page_commentary" level="2" n="115">
<p>
<s xml:id="echoid-s683" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s683" xml:space="preserve">
On this page Harriot explores 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><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>f</mi><mo maxsize="1">)</mo><mo>=</mo><mn>0</mn></mstyle></math>,
the conjugate of the second equation on Add MS 6783, 177. <lb/>
As there, he demonstrates the conditions required for the terms in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>, or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> to vanish.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head112" xml:space="preserve" xml:lang="lat">
d.8.) De generatione æquationum canonicarum
<lb/>[<emph style="it">tr: 
On the generation of canonical equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s685" xml:space="preserve">
Quatuor istæ sequentes <lb/>
æquationes sunt coniugatæ <lb/>
suis comparibus (d.7.)
<lb/>[<emph style="it">tr: 
These four following equations are conjugates, paired with those on d.7
</emph>]<lb/>
[<emph style="it">Note: 
Page d.7 is Add MS 6783, f. 178.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s686" xml:space="preserve">
Ex hac æquatione <lb/>
oriuntur tres <lb/>
sequentes.
<lb/>[<emph style="it">tr: 
From this equation arise the three following.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s687" 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><mo>=</mo><mi>f</mi></mstyle></math>. Tollitur tertius gradus, nempe (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>). <lb/>
et per reductionem, <lb/>
fit:
<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><mo>=</mo><mi>f</mi></mstyle></math>, the third degree term (that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>) may be removed,
whence by reduction there comes:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s688" xml:space="preserve">
Si: <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><mo>=</mo><mi>b</mi><mi>f</mi><mo>+</mo><mi>c</mi><mi>f</mi><mo>+</mo><mi>d</mi><mi>f</mi></mstyle></math>. Tollitur secundus gradus, nempe (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>). <lb/>
Et per reductionem, <lb/>
fit:
<lb/>[<emph style="it">tr: 
If <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><mo>=</mo><mi>b</mi><mi>f</mi><mo>+</mo><mi>c</mi><mi>f</mi><mo>+</mo><mi>d</mi><mi>f</mi></mstyle></math>, the second degree term (that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>) may be removed,
whence by reduction there comes:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s689" xml:space="preserve">
Si: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>d</mi><mo>=</mo><mi>b</mi><mi>c</mi><mi>f</mi><mo>+</mo><mi>b</mi><mi>d</mi><mi>f</mi><mo>+</mo><mi>c</mi><mi>d</mi><mi>f</mi></mstyle></math>. Tollitur primus gradus, nempe (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>). <lb/>
Et per reductionem, <lb/>
fit:
<lb/>[<emph style="it">tr: 
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>d</mi><mo>=</mo><mi>b</mi><mi>c</mi><mi>f</mi><mo>+</mo><mi>b</mi><mi>d</mi><mi>f</mi><mo>+</mo><mi>c</mi><mi>d</mi><mi>f</mi></mstyle></math>, the first degree term (that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>) may be removed,
whence by reduction there comes:
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f176v" o="176v" n="352"/>
<pb file="add_6783_f177" o="177" n="353"/>
<div xml:id="echoid-div116" type="page_commentary" level="2" n="116">
<p>
<s xml:id="echoid-s690" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s690" xml:space="preserve">
On this page Harriot explores 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><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>f</mi><mo maxsize="1">)</mo><mo>=</mo><mn>0</mn></mstyle></math> and <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><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>f</mi><mo maxsize="1">)</mo><mo>=</mo><mn>0</mn></mstyle></math>. <lb/>
In the latter case, he demonstrates the conditions required for the terms in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>, or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> to vanish,
and finds the resulting form of the coefficents in terms of the three positive roots <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/>
From this it is easy for him to see the close relationship between the equations in Add MS 6783, f. 181 and f. 180:
the positive root in the latter is the sum of the positive roots in the former.
In modern terms we sould say that the positive roots in one equation are the negative roots in the other.
Harriot calls such pairs 'conjugate' or 'converse'.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head113" xml:space="preserve" xml:lang="lat">
d.7.) De generatione æquationum canonicarum
<lb/>[<emph style="it">tr: 
On the generation of canonical equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s692" xml:space="preserve">
Ex hac æquatione <lb/>
oriuntur tres <lb/>
sequentes.
<lb/>[<emph style="it">tr: 
From this equation arise the three follwing.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s693" 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><mo>=</mo><mi>f</mi></mstyle></math>. Tollitur tertius gradus, nempe (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>). <lb/>
et per reductionem fit:
<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><mo>=</mo><mi>f</mi></mstyle></math>, the third degree term (that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>) may be removed.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s694" xml:space="preserve">
Si: <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><mo>=</mo><mi>b</mi><mi>f</mi><mo>+</mo><mi>c</mi><mi>f</mi><mo>+</mo><mi>d</mi><mi>f</mi></mstyle></math>. <lb/>
Tollitur secundus gradus, nempe (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>). <lb/>
unde per reductionem, <lb/>
fit:
<lb/>[<emph style="it">tr: 
If <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><mo>=</mo><mi>b</mi><mi>f</mi></mstyle></math>, the second degree term (that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>) may be removed,
whence by reduction there comes:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s695" xml:space="preserve">
Si: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>d</mi><mo>=</mo><mi>b</mi><mi>c</mi><mi>f</mi><mo>+</mo><mi>b</mi><mi>d</mi><mi>f</mi><mo>+</mo><mi>c</mi><mi>d</mi><mi>f</mi></mstyle></math>. <lb/>
Tollitur primus gradus, nempe (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>). <lb/>
unde per reductionem, <lb/>
fit:
<lb/>[<emph style="it">tr: 
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>d</mi><mo>=</mo><mi>b</mi><mi>c</mi><mi>f</mi><mo>+</mo><mi>b</mi><mi>d</mi><mi>f</mi><mo>+</mo><mi>c</mi><mi>d</mi><mi>f</mi></mstyle></math>, the first degree term (that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>) may be removed,
whence by reduction there comes:
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f177v" o="177v" n="354"/>
<pb file="add_6783_f178" o="178" n="355"/>
<div xml:id="echoid-div117" type="page_commentary" level="2" n="117">
<p>
<s xml:id="echoid-s696" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s696" xml:space="preserve">
In this page Harriot begins to explore the consequences of using quadratic factors of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>±</mo><mi>b</mi><mi>c</mi></mstyle></math>.
He describes the resulting equations as 'reciprocal equations'.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head114" xml:space="preserve" xml:lang="lat">
d.6.) De generatione æquationum canonicarum <lb/>
De reciprocis æquationibus.
<lb/>[<emph style="it">tr: 
On the generation of canonical equations <lb/>
On reciprocal equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s698" xml:space="preserve">
In solido. <lb/>[...]<lb/> sit
<lb/>[<emph style="it">tr: 
In the solid ... let
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s699" xml:space="preserve">
In solido. <lb/>[...]<lb/> sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>d</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
In the solid ... let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>d</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s700" xml:space="preserve">
Æquatio erit:
<lb/>[<emph style="it">tr: 
The equation will be:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s701" xml:space="preserve">
Consectarium
<lb/>[<emph style="it">tr: 
Consequence
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s702" xml:space="preserve">
Si coefficiens planum, multiplicatum coefficiente longitudine; <lb/>
faciat solidum datum: radix nota erit sine resolutione.
<lb/>[<emph style="it">tr: 
If the coefficient of the linear term is multiplied by the coefficient of the square term,
it makes the given solid: the root is then known without solving.
</emph>]<lb/>
[<emph style="it">Note: 
In Harriot's dimensionally consistent equations, the coefficient of the linear term must be regarded as 2-dimensional,
that is, as 'coefficiens planum'.
Similarly, the coefficient of the square term mut be regarded as 1=dimensional,
that is, as 'coefficens longitudinum'.
The 'solidum datum' is the given known term, which is regarded as 3-dimensional.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s703" xml:space="preserve">
In æquationibus omnino affirmatis ita se habet: <lb/>
sed de his et <lb/>
alijs huius generis <lb/>
alibi.
<lb/>[<emph style="it">tr: 
In any positive equation it is thus: but of these and others of this kind elsewhere.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s704" 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>6</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>9</mn><mi>a</mi><mo>=</mo><mn>5</mn><mn>4</mn></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> non <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mn>2</mn></mstyle></math>.
<lb/>[<emph style="it">tr: 
<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>9</mn><mi>a</mi><mo>=</mo><mn>5</mn><mn>4</mn></mstyle></math>; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> is not equal to 2.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s705" xml:space="preserve">
In his æquationibus: <lb/>
coefficiens planum <lb/>
multiplicatum coef-<lb/>
ficiente longitudine <lb/>
facit <lb/>
solidum datum.
<lb/>[<emph style="it">tr: 
In these equaions, the coefficient of the linear term multiplied by the coefficient of the square gives the given solid.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s706" xml:space="preserve">
Et: <lb/>
sub gradus faciunt <lb/>
potestatum.
<lb/>[<emph style="it">tr: 
And under the degree arise the powers.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s707" xml:space="preserve">
In æquationibus <lb/>
ipsis, habent reci-<lb/>
procum positivem <lb/>
ut:
<lb/>[<emph style="it">tr: 
In these equatons, the reciprocals are positive, as:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s708" xml:space="preserve">
et ideo <lb/>
apellentur <lb/>
reciprocæ
<lb/>[<emph style="it">tr: 
and therefore they are called reciprocals.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f178v" o="178v" n="356"/>
<pb file="add_6783_f179" o="179" n="357"/>
<div xml:id="echoid-div118" type="page_commentary" level="2" n="118">
<p>
<s xml:id="echoid-s709" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s709" xml:space="preserve">
On this page Harriot continues his exploration of the equation generated by 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><mo>=</mo><mn>0</mn></mstyle></math>. Here he demonstrates the conditions required for the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> to vanish,
and finds the resulting form of the coefficents in terms 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>. <lb/>
As in Add MS 6783, f. 180, he refers back to Add MS 6783, f. 181 to show that this process produces a pair
of conjugate equations, that is, the positive roots of one are the negative roots of the other.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head115" xml:space="preserve" xml:lang="lat">
d.5.) De generatione æquationum canonicarum
<lb/>[<emph style="it">tr: 
On the generation of canonical equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s711" xml:space="preserve">
In superiori æquatione, nempe:
<lb/>[<emph style="it">tr: 
In the above equation, that is:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s712" xml:space="preserve">
Si, <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>. Tollitur <lb/>
gradus primus, nempe (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>)
<lb/>[<emph style="it">tr: 
If <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>, the first degree term (that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>) may be removed.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s713" xml:space="preserve">
et ita est: <lb/>
est igitur.
<lb/>[<emph style="it">tr: 
And so it is. Therefore it is so.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s714" xml:space="preserve">
Hinc ex hac æquatione et altera (d.3) <lb/>
fiunt æquationes coniugatæ.
<lb/>[<emph style="it">tr: 
Here from this equation and others (d.3) arise conjugate equations.
</emph>]<lb/>
[<emph style="it">Note: 
Page d.3 is Add MS 6783, f. 180.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s715" xml:space="preserve">
Aliter.
<lb/>[<emph style="it">tr: 
Another way.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s716" xml:space="preserve">
si: <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>. Tollitur gradus <lb/>
primus, nempe (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>)
<lb/>[<emph style="it">tr: 
If <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>, the first degree term (that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>) may be removed.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s717" xml:space="preserve">
Et sunt coniugatæ
<lb/>[<emph style="it">tr: 
And they are conjugate.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s718" xml:space="preserve">
altera coniugata <lb/>
potest etiam fieri ex (d.3
<lb/>[<emph style="it">tr: 
Other conjugates may also arise from d.3.
</emph>]<lb/>
[<emph style="it">Note: 
Page d.3 is Add MS 6783, f. 180.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6783_f179v" o="179v" n="358"/>
<pb file="add_6783_f180" o="180" n="359"/>
<div xml:id="echoid-div119" type="page_commentary" level="2" n="119">
<p>
<s xml:id="echoid-s719" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s719" xml:space="preserve">
On this page Harriot explores 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><mo>=</mo><mn>0</mn></mstyle></math>.
He states that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is the only (positive) roots of this equation but does not offer any justification. <lb/>
As in Add MS 6783, f. 181, he demonstrates the conditions required for the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> to vanish,
and finds the resulting form of the coefficents in terms 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>. <lb/>
From this it is easy for him to see the close relationship between the equations in Add MS 6783, f. 181 and f. 180:
the positive root in the latter is the sum of the positive roots in the former.
In modern terms we sould say that the positive roots in one equation are the negative roots in the other.
Harriot calls such pairs 'conjugate' or 'converse'.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head116" xml:space="preserve" xml:lang="lat">
d.4.) De generatione æquationum canonicarum
<lb/>[<emph style="it">tr: 
On the generation of canonical equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s721" xml:space="preserve">
In solido. <lb/>[...]<lb/> sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>d</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
In the product <lb/>[...]<lb/> let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>d</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s722" xml:space="preserve">
et erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>d</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
and then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>d</mi></mstyle></math>
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s723" xml:space="preserve">
non <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>b</mi></mstyle></math> vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> <lb/>
neque alteri præter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
not <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 anything other than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s724" xml:space="preserve">
Analytice probatur ut supra.
<lb/>[<emph style="it">tr: 
Proved analytically above.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s725" xml:space="preserve">
In superiori æquatione:
<lb/>[<emph style="it">tr: 
In the above equation:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s726" xml:space="preserve">
Si, <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>. tollitur secundus <lb/>
gradus nempe (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>)
<lb/>[<emph style="it">tr: 
If <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>, the term of second degree (that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>) may be removed.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s727" xml:space="preserve">
Si: <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/>
erit:
<lb/>[<emph style="it">tr: 
If <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> then:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s728" xml:space="preserve">
est enim: <lb/>
est igitur.
<lb/>[<emph style="it">tr: 
Indeed it is. Therefore it is so.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s729" xml:space="preserve">
Hinc ex hac æquatione et altera pag. d.3. <lb/>
oritur insigne poristicum:
<lb/>[<emph style="it">tr: 
Here from this equation and others on page d.3 there arises this notable property:
</emph>]<lb/>
[<emph style="it">Note: 
Page d.3 is Add MS 6783, f. 181.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s730" xml:space="preserve">
sunt enim æquationes coniugatæ <lb/>
<emph style="st">sive conversæ</emph>
<lb/>[<emph style="it">tr: 
For the equations are conjugate.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s731" xml:space="preserve">
Aliter.
<lb/>[<emph style="it">tr: 
Another way.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s732" xml:space="preserve">
In superiori æquatione:
<lb/>[<emph style="it">tr: 
In the above equation:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s733" xml:space="preserve">
Sed si sumatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>&gt;</mo><mi>d</mi></mstyle></math> <lb/>
erit æquatio conversa, ita:
<lb/>[<emph style="it">tr: 
But if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is taken to be greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> then we have the converse equation, thus:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s734" xml:space="preserve">
possunt ita disponi utraque <lb/>
et habentur dupliciter coniugata.
<lb/>[<emph style="it">tr: 
They may thus be displayed either way, and are doubly conjugate.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s735" xml:space="preserve">
Probantur.
<lb/>[<emph style="it">tr: 
Proved.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f180v" o="180v" n="360"/>
<pb file="add_6783_f181" o="181" n="361"/>
<div xml:id="echoid-div120" type="page_commentary" level="2" n="120">
<p>
<s xml:id="echoid-s736" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s736" xml:space="preserve">
On this page Harriot explores 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><mo>=</mo><mn>0</mn></mstyle></math>.
He argues that <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 (positive) roots of this equation,
and that any other root <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> can only be equal to either <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>. <lb/>
Equations with two positive roots are precisely the kind Harriot is interested in for the purposes of Section c.
He therefore demonstrates the conditions required for the terms in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> or in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> to vanish,
and the form of the resulting coefficents in terms 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>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head117" xml:space="preserve" xml:lang="lat">
d.3.) De generatione æquationum canonicarum
<lb/>[<emph style="it">tr: 
On the generation of canonical equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s738" xml:space="preserve">
In solido. <lb/>[...]<lb/> sit <lb/>[...]<lb/>
<lb/>[<emph style="it">tr: 
In the solid ... let
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s739" xml:space="preserve">
Et non, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>d</mi></mstyle></math> <lb/>
neque <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> alteri præter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
And <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> is not equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, nor to any <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> besides <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>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s740" xml:space="preserve">
Non est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>d</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> is not equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s741" xml:space="preserve">
Si sit:
<lb/>[<emph style="it">tr: 
If it were:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s742" xml:space="preserve">
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mi>d</mi></mstyle></math>. contra propositionem.
<lb/>[<emph style="it">tr: 
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mi>d</mi></mstyle></math>, against the proposition.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s743" xml:space="preserve">
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mi>d</mi></mstyle></math>. contra propositionem.
<lb/>[<emph style="it">tr: 
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mi>d</mi></mstyle></math>, against the proposition.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s744" xml:space="preserve">
vel aliter.
<lb/>[<emph style="it">tr: 
or otherwise
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s745" xml:space="preserve">
si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>=</mo><mi>b</mi></mstyle></math>. vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. esset æqualitas <lb/>
ut apparet.
<lb/>[<emph style="it">tr: 
if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>=</mo><mi>b</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>=</mo><mi>c</mi></mstyle></math> there would be equality, as is clear.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s746" xml:space="preserve">
sed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> non <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>b</mi></mstyle></math>, aut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
but <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is not equal to <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>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s747" xml:space="preserve">
ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> non <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>a</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is not equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s748" xml:space="preserve">
Et simile ratione ut supra non erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>=</mo><mi>a</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
And by similar reasoning to the above, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> is not equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s749" xml:space="preserve">
Ergo.
<lb/>[<emph style="it">tr: 
Therefore.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s750" xml:space="preserve">
Sit ut supra
<lb/>[<emph style="it">tr: 
Suppose as above
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s751" xml:space="preserve">
Si <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>. Tollitur <lb/>
secundus graus, nempe (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>)
<lb/>[<emph style="it">tr: 
If <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>, the term of second degree (that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>) may be removed.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s752" xml:space="preserve">
analytica <lb/>
demonstratio ut <lb/>
supra. et (c.3)
<lb/>[<emph style="it">tr: 
Demonstrated analytically as above and in (c.3).
</emph>]<lb/>
[<emph style="it">Note: 
Sheet c.3 is Add MS 6782, f. 415.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s753" xml:space="preserve">
Sit ut supra
<lb/>[<emph style="it">tr: 
Suppose as above
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s754" xml:space="preserve">
Si <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>. Tollitur primus <lb/>
gradus, nempe (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>)
<lb/>[<emph style="it">tr: 
If <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>, the term of first degree (that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>) may be removed.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s755" xml:space="preserve">
Analytica demonstratio (c. <emph style="st">4</emph> <emph style="super">7</emph>)
<lb/>[<emph style="it">tr: 
Demonstrated analytically in (c.7)
</emph>]<lb/>
[<emph style="it">Note: 
Sheet c.7 is Add MS 6782, f. 411.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6783_f181v" o="181v" n="362"/>
<pb file="add_6783_f182" o="182" n="363"/>
<div xml:id="echoid-div121" type="page_commentary" level="2" n="121">
<p>
<s xml:id="echoid-s756" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s756" xml:space="preserve">
On this page Harriot explores 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><mo>=</mo><mn>0</mn></mstyle></math>.
He argues that <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> are all roots of this equation,
but that there can be no other root <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, say.
His argument here is less convincing, however, than the simpler argument for quadratics on Add MS 6783, f. 183.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head118" xml:space="preserve" xml:lang="lat">
d.2.) De generatione æquationum canonicarum
<lb/>[<emph style="it">tr: 
On the generation of canonical equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s758" xml:space="preserve">
In solido. <lb/>[...]<lb/> vel <lb/>[...]<lb/> sit <lb/>[...]<lb/>
<lb/>[<emph style="it">tr: 
In the solid ... or ... let
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s759" xml:space="preserve">
Et non <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> alteri præter <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>.
<lb/>[<emph style="it">tr: 
And not some <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> other than <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>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s760" xml:space="preserve">
si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math> <lb/>
erit: <lb/>
et ita est.
<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 <lb/>
and it is so.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s761" 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/>
et ita est.
<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 <lb/>
and it is so.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s762" xml:space="preserve">
si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>d</mi></mstyle></math> <lb/>
erit: <lb/>
est enim. est igitur.
<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 <lb/>
and indeed it is. Therefore it is so.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s763" xml:space="preserve">
Non erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>f</mi></mstyle></math>. alteri præter, <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>.
<lb/>[<emph style="it">tr: 
Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> is not equal to any <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> other than <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>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s764" xml:space="preserve">
si sit:
<lb/>[<emph style="it">tr: 
If it were:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s765" xml:space="preserve">
si sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>=</mo><mi>b</mi></mstyle></math> vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>: vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>: <lb/>
esset æqualitas, ut apparet.
<lb/>[<emph style="it">tr: 
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> were equal to <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>d</mi></mstyle></math>, then there would be equality, as is clear.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s766" xml:space="preserve">
Sed non sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>=</mo><mi>b</mi></mstyle></math> vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>: vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>: <lb/>
ex suppositione.
<lb/>[<emph style="it">tr: 
But <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> is not <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>d</mi></mstyle></math> by supposition.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s767" xml:space="preserve">
ergo non est. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>=</mo><mi>a</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
Therefore it is not the case that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>=</mo><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f182v" o="182v" n="364"/>
<pb file="add_6783_f183" o="183" n="365"/>
<div xml:id="echoid-div122" type="page_commentary" level="2" n="122">
<p>
<s xml:id="echoid-s768" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s768" xml:space="preserve">
In this 21-page section (Section d), Harriot developes one of his most original ideas in algebra:
the generation of polynomial equations from the multiplication of linear and quadratic factors.
All previous treatments of quadratic, cubic, or quartic equations had been based on traditional ideas of proportion.
Harriot broke decisively away from such a model, completely changing the way equations were perceived and studied.
This was the one section of Harriot's work that was published after his death,
edited by his friend Walter Warner as the <emph style="it">Artis analyticae praxis</emph> (1631).
Warner, however, seems not to have fully understood all of Harriot's working, and the version given in the
<emph style="it">Praxis</emph> is very different from the systematic treatment seen in the mansucripts. Further,
Warner eliminated all negative or imaginary roots, leading later readers to assume that Harriot never handled them.
That he did so is clear from the later pages in this section. <lb/>
The 'canoncial equations' in Section d are prototypes, generated from a given set of roots.
Note that Harriot has already used these canonical equations in Section c (Add MS 6783, f. 417 to f. 400);
see, for example, Add MS 6783, f. 417, f (for quadratics),
f. 415, f. 414, f. 411, f. 410, f. 409, f. 408 (for cubics),
f. 407, f. 406, f. 404, f. 403, f. 402, f. 400 (for quartics).
These are different from some other equations, also described as 'canonical',
also to be found in Section c. For a page that shows both types, see Add MS 6782, f. 408.
There the first canonical form is of the kind derived here in Section d,
for a cubic equation with no linear term, and with 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>.
The second canonical form is used for numerical solution;
in that 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> represent the first and second digits of the root.
Thus <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> have entirely different interpretations in the two forms;
fortunately, their specific meanings in any calculation are always clear from the context. <lb/>
The equations in Section c are all trinomial in form,
with the highest power subtracted (torn away, or 'avulsed') from the second highest power.
Such equations have either no positive roots or two of them.
From his study of Viète's numerical method, Harriot was particularly intersted in the latter case.
His aim in Section d is to discover the conditions under which an equation with two postive roots
reduces to trinomial form, and the nature of the resulting coefficients in terms of those two roots.
Thus Sections c and d are very closely related, and there are several specific cross-references from one to the other.
For further discussion see Stedall 2011. <lb/>
As for other parts of Harriot's work on equations,
algebraic manipulations and single words contained in them are not transcribed in this section.
<lb/>
<lb/>
In this first page (d.1) Harriot sets out the multiplications he intends to explore, in modern notation
<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>, <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></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>b</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>, and so on. <lb/>
After the trivial case <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>b</mi><mo>=</mo><mn>0</mn></mstyle></math>, Harriot treats 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>=</mo><mn>0</mn></mstyle></math>,
arguing that <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 both (positive) roots of this equation,
but that there can be no other (positive) root <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, say. <lb/>
He then treats 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></mstyle></math> in a similar way,
arguing that in this case <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math> is the only (positive) root.
At this stage he is still concerned only with positive roots, though this will change later.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head119" xml:space="preserve" xml:lang="lat">
 d.1.) De generatione æquationum canonicarum
<lb/>[<emph style="it">tr: 
On the generation of canonical equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s770" xml:space="preserve">
Methodus earum <lb/>
quæ sequentur.
<lb/>[<emph style="it">tr: 
The way of proceeding for those that follow.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s771" xml:space="preserve">
In longo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>b</mi></mstyle></math> <lb/>
vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>a</mi></mstyle></math> <lb/>
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
In the linear term <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>, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s772" xml:space="preserve">
ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>b</mi><mo>=</mo><mn>0</mn></mstyle></math> <lb/>
vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>a</mi><mo>=</mo><mn>0</mn></mstyle></math>.
<lb/>[<emph style="it">tr: 
therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>b</mi><mo>=</mo><mn>0</mn></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>a</mi><mo>=</mo><mn>0</mn></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s773" xml:space="preserve">
ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math> <emph style="st">neque</emph> <emph style="super">et non</emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> <lb/>
alteri præter <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>a</mi><mo>=</mo><mi>b</mi></mstyle></math> and not <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s774" xml:space="preserve">
si sit: <lb/>
erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>=</mo><mi>b</mi></mstyle></math> <lb/>
contra positionem <lb/>
nam ponitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> alterum <lb/>
quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
If it were, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>=</mo><mi>b</mi></mstyle></math>, against what was laid down, for it is supposed that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is other than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s775" xml:space="preserve">
Sit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math> <lb/>
et: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math> <lb/>
In <emph style="st">rectangulo</emph> plano
<lb/>[<emph style="it">tr: 
Let <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> in the product
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s776" xml:space="preserve">
Etsi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> inæquales.
<lb/>[<emph style="it">tr: 
Although <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 unequal.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s777" xml:space="preserve">
et non altera præter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
And none other besides <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>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s778" xml:space="preserve">
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> <lb/>
et ita est:
<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>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-s779" 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> <lb/>
et ita est:
<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>; and so it is.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s780" xml:space="preserve">
Est enim. est igitur.
<lb/>[<emph style="it">tr: 
Indeed it is. Therefore it is so.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s781" xml:space="preserve">
* Neque
<lb/>[<emph style="it">tr: 
Not
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s782" xml:space="preserve">
Sit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math> <lb/>
In <emph style="st">rectangulo</emph> plano
<lb/>[<emph style="it">tr: 
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math> in the product.
</emph>]<lb/>
</s>
<lb/>
<lb/>
<s xml:id="echoid-s783" xml:space="preserve">
erit <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></mstyle></math> non æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. neque alteri præter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> is not equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, nor anything else besides <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s784" 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>c</mi><mo>=</mo><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: 
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><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-s785" xml:space="preserve">
ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mi>c</mi></mstyle></math> contra positionem.
<lb/>[<emph style="it">tr: 
therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mi>c</mi></mstyle></math>, against what was laid down.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s786" xml:space="preserve">
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>. et non <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>c</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math> and not <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-s787" xml:space="preserve">
Neque erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>d</mi></mstyle></math> alteri præter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
Nor is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, or anything else besides <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s788" xml:space="preserve">
si sit:
</s>
<lb/>
<s xml:id="echoid-s789" xml:space="preserve">
ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mi>d</mi></mstyle></math> contra positionem. <lb/>
nam ponitur altera quam <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><mo>=</mo><mi>d</mi></mstyle></math> against what was laid down, for it was supposed other than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s790" xml:space="preserve">
Si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mi>c</mi></mstyle></math>. tollitur gradus <lb/>
primus
<lb/>[<emph style="it">tr: 
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mi>c</mi></mstyle></math> the term of first degree may be removed.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s791" xml:space="preserve">
et erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>=</mo><mi>a</mi><mi>a</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
and then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>=</mo><mi>a</mi><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s792" xml:space="preserve">
et: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s793" xml:space="preserve">
* Neque erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>d</mi></mstyle></math> alteri præter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
</s>
<lb/>
<s xml:id="echoid-s794" xml:space="preserve">
si sit:
</s>
<lb/>
<s xml:id="echoid-s795" xml:space="preserve">
ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mi>d</mi></mstyle></math> contra positionem. <lb/>
</s>
<lb/>
<s xml:id="echoid-s796" xml:space="preserve">
ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mi>d</mi></mstyle></math> contra positionem. <lb/>
</s>
</p>
<pb file="add_6783_f183v" o="183v" n="366"/>
<pb file="add_6783_f184" o="184" n="367"/>
<div xml:id="echoid-div123" type="page_commentary" level="2" n="123">
<p>
<s xml:id="echoid-s797" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s797" xml:space="preserve">
This page contains the proofs of Lemma 3, required to support condition (ii) on page e. 27 (Add MS 6783, f. 186).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head120" xml:space="preserve" xml:lang="lat">
e.29.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s799" xml:space="preserve">
Appendicula.
<lb/>[<emph style="it">tr: 
A short appendix
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s800" xml:space="preserve">
3. lemma, et poristicum.
<lb/>[<emph style="it">tr: 
Lemma 3, and rule.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s801" xml:space="preserve">
Si magnitudo secetur in tres inæquales partes: cubus <lb/>
e tertie parte, totius maius est solido e tribus inæqualibus.
<lb/>[<emph style="it">tr: 
If a magnitude is cut into three unequal parts,
the cube of a third of the total is greater than the product of the three unequal parts.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s802" xml:space="preserve">
per lem: 2. <lb/>
(e.8.)
<lb/>[<emph style="it">tr: 
by lemma 2 (e.8)
</emph>]<lb/>
[<emph style="it">Note: 
Lemma 2 on sheet e.8 is Add MS 6783, f. 106v.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s803" xml:space="preserve">
per lem: 1. <lb/>
(e.8.)
<lb/>[<emph style="it">tr: 
by lemma 1 (e.8)
</emph>]<lb/>
[<emph style="it">Note: 
Lemma 1 on sheet e.8 is on Add MS 6783, f. 106v.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s804" xml:space="preserve">
Est igitur:. <lb/>
Quod demonstrandum.
<lb/>[<emph style="it">tr: 
Therefore it is so; as was to be demonstrated.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f184v" o="184v" n="368"/>
<pb file="add_6783_f185" o="185" n="369"/>
<div xml:id="echoid-div124" type="page_commentary" level="2" n="124">
<p>
<s xml:id="echoid-s805" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s805" xml:space="preserve">
This page contains the proofs of Lemma 2, required to support condition (i) on page e. 27 (Add MS 6783, f. 186).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head121" xml:space="preserve" xml:lang="lat">
e.28.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s807" xml:space="preserve">
Appendicula.
<lb/>[<emph style="it">tr: 
A short appendix
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s808" xml:space="preserve">
2. lemma, et poristicum.
<lb/>[<emph style="it">tr: 
Lemma 2, and rule.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s809" xml:space="preserve">
Si magnitudo secetur in tres inæquales partes: triplex <lb/>
quadratum e tertie parte totius maius est tribus planis <lb/>
e <emph style="super">singulis</emph> binis inæqualibus.
<lb/>[<emph style="it">tr: 
If a magnitude is cut into three unequal parts,
three times the square of a third of the total is more than the three products from each unequal pair.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s810" xml:space="preserve">
per lem: 1. <lb/>
(e.8.)
<lb/>[<emph style="it">tr: 
by lemma 1 (e.8)
</emph>]<lb/>
[<emph style="it">Note: 
Lemma 1 on sheet e.8 is on Add MS 6783, f. 106v.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s811" xml:space="preserve">
Est igitur:. <lb/>
Quod demonstrandum.
<lb/>[<emph style="it">tr: 
Therefore it is so; as was to be demonstrated.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f185v" o="185v" n="370"/>
<pb file="add_6783_f186" o="186" n="371"/>
<div xml:id="echoid-div125" type="page_commentary" level="2" n="125">
<p>
<s xml:id="echoid-s812" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s812" xml:space="preserve">
This page begins with the sixth and last of the equations given by Viète in
<emph style="it">De numerosa potestum resolutione</emph>, in the section entitled
'De ambiguitate cubi multipliciter adfecti' ('On ambiguity in cubic equations with multiple terms'),
following Problem 18.  <lb/>
Harriot then goes on to discuss some of the 'precepts' given by Viète, with his own corrections to them. <lb/>
Viète's first equation was (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>1</mn><mn>1</mn><mi>x</mi><mo>=</mo><mn>6</mn></mstyle></math>.
Harriot gives this in the form <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>.
Viète claimed that this has three real and distinct roots (namely, 1, 2, and 3) because
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mo>×</mo><mrow><msup><mrow><mo>(</mo><mfrac><mrow><mn>6</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow><mo>&gt;</mo><mn>1</mn><mn>1</mn></mstyle></math>.
In more general terms, for an equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>-</mo><mi>p</mi><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>+</mo><mi>q</mi><mi>x</mi><mo>-</mo><mi>r</mi><mo>=</mo><mn>0</mn></mstyle></math>, this condition may be written
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mrow><msup><mrow><mo>(</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow><mo>&gt;</mo><mi>q</mi></mstyle></math> (condition (i)).
It is true that if all the roots are real, this condition will be satisfied.
If the roots are <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>, for instance, then it is equivalent to
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mrow><msup><mrow><mo>(</mo><mfrac><mrow><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow><mo>&gt;</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>d</mi><mo>+</mo><mi>d</mi><mi>b</mi></mstyle></math>,
precisely the inequality Harriot proves on the next page, sheet e.28 (Add MS 6783, 185), Lemma 2.
However, although this condition is necessary for the equation to have three real roots, it is not sufficient.
To demonstrate this, Harriot adjoins a second example, <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>1</mn><mn>2</mn></mstyle></math>,
where the condition is again satisfied, but the equation has only one real root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>4</mn></mstyle></math>.
Harriot claims that a further condition is necessary for three (real) roots, namely,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mrow><mo>(</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow><mn>3</mn></msup></mrow><mo>&gt;</mo><mi>r</mi></mstyle></math> (condition (ii)).
This is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mrow><mo>(</mo><mfrac><mrow><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow><mn>3</mn></msup></mrow><mo>&gt;</mo><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>,
the inequality that he proves on sheet e.29 (Add MS 6783, 184), Lemma 3.
Harriot's first equation satisfies condition (ii), but the second does not.
However, as before, he has only shown that condition (ii) is necessary for three real roots,
not that it is sufficient. <lb/>
Viète made a further claim: that the roots of the first equation (namely, 1, 2, and 3)
are equally spaced because <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mo>+</mo><mn>2</mn><mo>×</mo><mrow><msup><mrow><mo>(</mo><mfrac><mrow><mn>6</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow><mn>3</mn></msup></mrow><mo>=</mo><mn>1</mn><mn>1</mn><mo>×</mo><mfrac><mrow><mn>6</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math>.
In more general terms, using the notation above,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>+</mo><mn>2</mn><mrow><msup><mrow><mo>(</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow><mn>3</mn></msup></mrow><mo>=</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mi>q</mi></mstyle></math>.
Viète did not justify this statement but it is not difficult to check that this condition must indeed hold
if the roots are equally spaced. <lb/>
Harriot next turns to examples when condition (i) becomes an equality, that is,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mrow><msup><mrow><mo>(</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow><mo>=</mo><mi>r</mi></mstyle></math> (condition (i')).
Viète claimed that this leads to three equal roots, as in the equation (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>1</mn><mn>2</mn><mi>x</mi><mo>=</mo><mn>8</mn></mstyle></math>, with roots 2, 2, 2.
Harriot writes this in the form <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>.
Harriot again shows that although Viète's condition is necessary for three real roots, it is not sufficient.
He does so by adjoining a second example, <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>9</mn></mstyle></math>, where condition (i') is satisfied,
but the equation has only one real root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn></mstyle></math>.
Harriot claims that a further condition is necessary for three (real) roots, namely (in the above notation),
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>+</mo><mn>2</mn><mrow><msup><mrow><mo>(</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow><mn>3</mn></msup></mrow><mo>=</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mi>q</mi></mstyle></math> (condition (ii')).
This is precisely Viète's condtion for equally spaced roots,
but Harriot offers no justification for his statement. <lb/>
His first equation satisfies condition (ii'), but his second does not. <lb/>/
Harriot's third condition, in the above notation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mrow><msup><mrow><mo>(</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow><mo>&lt;</mo><mi>q</mi></mstyle></math>,
is the negation of condition (i);
when this holds there can be only one real root, as in his example, <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>3</mn><mi>a</mi><mo>=</mo><mn>8</mn></mstyle></math>. <lb/>
Harrot's fourth condition, in the above notation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mrow><mo>(</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow><mn>3</mn></msup></mrow><mo>&lt;</mo><mi>r</mi></mstyle></math>,
is the negation of condition (ii);
when this holds there can be only one real root, as in his two examples,
<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>1</mn><mn>2</mn></mstyle></math> and <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>9</mn></mstyle></math>. <lb/>
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head122" xml:space="preserve" xml:lang="lat">
e.27.) De resolutione æquationum per <emph style="st">æquationum</emph> reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s814" xml:space="preserve">
Appendicula.
<lb/>[<emph style="it">tr: 
A short appendix
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s815" xml:space="preserve">
est æquatio reciproca
<lb/>[<emph style="it">tr: 
which is a reciprocal equation
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s816" xml:space="preserve">
ideo solutio fit per compendia (d.6.) <lb/>
videlicet: <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo></mstyle></math> coeffcienti longitudini. 6.
<lb/>[<emph style="it">tr: 
therefore the solution can be found by a short cut from sheet d.6, namely,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> is the longitudinal coefficient, 6.
</emph>]<lb/>
[<emph style="it">Note: 
Sheet d.6 is Add MS 6783, f. 178.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s817" xml:space="preserve">
Vietæ pæcepta emendantur
<lb/>[<emph style="it">tr: 
Viète's precepts amended
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s818" xml:space="preserve">
* partis coefficientis <lb/>
plani, sit maior <lb/>
solido dato homoge-<lb/>
neo.
<lb/>[<emph style="it">tr: 
part of the plane coefficient, is greater than the given solid.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s819" xml:space="preserve">
cubus adfectus sub quadrato negate et latero affirmate <lb/>
ambiguus esset, quando triplum quadratum e triente coefficientis <lb/>
longitudinis maius est coefficiente plano [et cubus e triente <lb/>
eodem maior sit solido dato homgeneo.] et cubus lateris tertiæ <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>6</mn><mo>,</mo><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>1</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>6</mn></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>2</mn><mo>.</mo><mn>3</mn></mstyle></math> (e.26.) <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>6</mn><mo>,</mo><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>1</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>1</mn><mn>2</mn></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>4</mn></mstyle></math> (e.24) <lb/>
*
<lb/>[<emph style="it">tr: 
a cubic equation affected by a negtive square and a positive side will have more than one root
when three times the square of a third of the longitudinal coefficient is greater than the plane coefficient
(and the cube of the same third is greater than the given solid)
and is a cube with three sides. <lb/>
<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>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mstyle></math> (e.26.) <lb/>
<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>1</mn><mn>2</mn></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>4</mn></mstyle></math> (e.24) <lb/>
*
</emph>]<lb/>
[<emph style="it">Note: 
Sheet e.24 is Add MS 6783, f. 189; sheet e.26 is Add MS 6783, f. 187.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s820" xml:space="preserve">
At cum triplum quadratum e triente coefficientis longitudinis <lb/>
æquale est coefficenti plano [et datum solidum homogeneum plus <lb/>
duplo cubo ex eodem triente, est æquale solido e triente dicto <lb/>
et coeffciente plano]: sunt tria latera sed æqualia singula. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>6</mn><mo>,</mo><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>2</mn><mo>,</mo><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><mo>.</mo><mn>2</mn><mo>.</mo><mn>2</mn></mstyle></math> (e.26.) <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>6</mn><mo>,</mo><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>2</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>9</mn></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn></mstyle></math>
<lb/>[<emph style="it">tr: 
And when three times the square of a third of the longitudinal coefficient is equal to the plane coefficient
(and the given solid plus twice the cube of the same third
is equal to the product of the said third and the plane coefficient),
then there are three sides but all equal. <lb/>
<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>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>2</mn></mstyle></math> (e.26.) <lb/>
<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>9</mn></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn></mstyle></math>
</emph>]<lb/>
[<emph style="it">Note: 
Sheet e.26 is Add MS 6783, f. 187.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s821" xml:space="preserve">
Cum triplum quadratum [e triente] coefficientis longitu-<lb/>
dinis cedet coefficenti plano, nulla erit in radice ambi-<lb/>
guitas. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>6</mn><mo>,</mo><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>3</mn><mo>,</mo><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>1</mn></mstyle></math> (e.24.)
<lb/>[<emph style="it">tr: 
When three times the square (of a third) of the longitudinal coefficient is less than the plane coefficient,
there is no ambiguity in the roots. <lb/>
<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>3</mn><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>1</mn></mstyle></math> (e.24.)
</emph>]<lb/>
[<emph style="it">Note: 
Sheet e.24 is Add MS 6783, f. 189.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s822" xml:space="preserve">
Vel cum cubus [e triente] coefficientis longitudinis cedet <lb/>
solido dato homogeno. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>6</mn><mo>,</mo><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>1</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>1</mn><mn>2</mn></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>4</mn></mstyle></math> (e.24.) <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>6</mn><mo>,</mo><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>2</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>9</mn></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn></mstyle></math>
<lb/>[<emph style="it">tr: 
Or when the cube (of a third) of the longitudinal coefficient is less than the given solid. <lb/>
<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><mi>a</mi><mo>=</mo><mn>1</mn><mn>2</mn></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>4</mn></mstyle></math> (e.24.) <lb/>
<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>9</mn></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn></mstyle></math>
</emph>]<lb/>
[<emph style="it">Note: 
Sheet e.24 is Add MS 6783, f. 189.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s823" xml:space="preserve">
Sed quid opus est verbosis præceptis, cum species reductionis nostræ, non <lb/>
solum æquationis huius differentiæ sed cuiuslibet alterius latera omnia <lb/>
directe exhibat.
<lb/>[<emph style="it">tr: 
But what need is there fore verbose precepts, when the forms obtained by our reductions show immediately
not only all the roots of these cases of equations but of any others one wants.
</emph>]<lb/>
</s>
<s xml:id="echoid-s824" xml:space="preserve">
Attamen si harum præceptorum demonstratis <lb/>
desideratur, adposuimus tria lemmata sequentia.
<lb/>[<emph style="it">tr: 
Nevertheless, in case the demonstration of these precepts is wanted, we add the three following lemmas.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s825" xml:space="preserve">
1. Lemma. Canon trium laterum. (d.2.) (d.15.) (d.16.)
<lb/>[<emph style="it">tr: 
Lemma 1. The canonical form for three roots.
</emph>]<lb/>
[<emph style="it">Note: 
Sheets d.2, d.15, d.16 are Add MS 6783, f. 182, f. 169, f. 168, respectively.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6783_f186v" o="186v" n="372"/>
<pb file="add_6783_f187" o="187" n="373"/>
<div xml:id="echoid-div126" type="page_commentary" level="2" n="126">
<p>
<s xml:id="echoid-s826" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s826" xml:space="preserve">
The equations on this page are the first five of those discussed by Viète in
<emph style="it">De numerosa potestum resolutione</emph>, in a section entitled
'De ambiguitate cubi multipliciter adfecti' ('On ambiguity in cubic equations with multiple terms'),
following Problem 18. The sixth and last equation appears on the following page, e.27 (Add MS 6783, f. 186). <lb/>
Viète gave various conditions, all expressed verbally, and all offered without proof,
for such equations to have distinct or equal roots and, in the case of distinct roots,
for them to be equally or unequally spaced.
Harriot discusses these conditions, which he calls 'precepts', on the following page, e.27 (Add MS 6783, f. 186).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head123" xml:space="preserve" xml:lang="lat">
e.26.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s828" xml:space="preserve">
Appendicula ad intelligendum Vietæ locum de ambiguitate <lb/>
cubi multiplicitur adfecti. lib. de numerosa potestum resol: pag. 31.
<lb/>[<emph style="it">tr: 
A short appendix, on understanding that place in Viète on double roots of affected cubic equations,
<emph style="it">De numerosa potestatum resolutione</emph>, page 31.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s829" xml:space="preserve">
Exempla explicantur. Præcepta emendatur. Lemmata <lb/>
adponuntur et demonstrantur.
<lb/>[<emph style="it">tr: 
The examples explained; the precepts amended; the lemmas set out and demonstrated.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s830" xml:space="preserve">
parabalica
<lb/>[<emph style="it">tr: 
parabolic
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f187v" o="187v" n="374"/>
<pb file="add_6783_f188" o="188" n="375"/>
<div xml:id="echoid-div127" type="page_commentary" level="2" n="127">
<p>
<s xml:id="echoid-s831" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s831" xml:space="preserve">
The square term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mi>p</mi><mi>p</mi><mi>a</mi><mo>-</mo><mn>3</mn><mi>r</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>=</mo><mn>2</mn><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math> is removed by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mo>-</mo><mi>e</mi><mo>-</mo><mi>r</mi></mstyle></math>.
Similarly, the square term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>3</mn><mi>r</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>p</mi><mi>p</mi><mi>a</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math> is removed by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>e</mi><mo>-</mo><mi>r</mi></mstyle></math>. <lb/>
Conjugate pairs A and B, and C and D, are marked by lines from one column to another.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head124" xml:space="preserve" xml:lang="lat">
e.25.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s833" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda.
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f188v" o="188v" n="376"/>
<pb file="add_6783_f189" o="189" n="377"/>
<div xml:id="echoid-div128" type="page_commentary" level="2" n="128">
<p>
<s xml:id="echoid-s834" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s834" xml:space="preserve">
The square term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>3</mn><mi>r</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>p</mi><mi>p</mi><mi>a</mi><mo>=</mo><mn>2</mn><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math> is removed by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>e</mi><mo>+</mo><mi>r</mi></mstyle></math>.
Similarly, the square term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mi>p</mi><mi>p</mi><mi>a</mi><mo>+</mo><mn>3</mn><mi>r</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math> is removed by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>r</mi><mo>-</mo><mi>e</mi></mstyle></math>. <lb/>
Conjugate pairs A and B, and C and D, are marked by lines from one column to another. <lb/>
Below the working are two numerical examples of elliptic equations solved for all three roots,
and two hyperbolic equations solved for their single positive roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head125" xml:space="preserve" xml:lang="lat">
e.24.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s836" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda.
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s837" xml:space="preserve">
Hyperbolica <lb/>
æquatio.
<lb/>[<emph style="it">tr: 
A hyperbolic equation
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f189v" o="189v" n="378"/>
<pb file="add_6783_f190" o="190" n="379"/>
<div xml:id="echoid-div129" type="page_commentary" level="2" n="129">
<p>
<s xml:id="echoid-s838" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s838" xml:space="preserve">
The square term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo><mi>p</mi><mi>p</mi><mi>a</mi><mo>-</mo><mn>3</mn><mi>r</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>=</mo><mn>2</mn><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math> is removed by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mo>-</mo><mi>r</mi><mo>-</mo><mi>e</mi></mstyle></math>.
Similarly, the square term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mi>p</mi><mi>p</mi><mi>a</mi><mo>+</mo><mn>3</mn><mi>r</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math> is removed by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>e</mi><mo>-</mo><mi>r</mi></mstyle></math>. <lb/>
Conjugate pairs A and B, and C and D, are marked by lines from one column to another. <lb/>
Below the working is a numerical example of an elliptic equation solved for all three roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head126" xml:space="preserve" xml:lang="lat">
e.23.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s840" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda.
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f190v" o="190v" n="380"/>
<pb file="add_6783_f191" o="191" n="381"/>
<div xml:id="echoid-div130" type="page_commentary" level="2" n="130">
<p>
<s xml:id="echoid-s841" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s841" xml:space="preserve">
The square term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mi>p</mi><mi>p</mi><mi>a</mi><mo>-</mo><mn>3</mn><mi>r</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>=</mo><mn>2</mn><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math> is removed by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>e</mi><mo>+</mo><mi>r</mi></mstyle></math>.
Similarly, the square term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo><mi>p</mi><mi>p</mi><mi>a</mi><mo>+</mo><mn>3</mn><mi>r</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math> is removed by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>r</mi><mo>-</mo><mi>e</mi></mstyle></math>. <lb/>
Conjugate pairs A and B, and C and D, are marked by lines from one column to another. <lb/>
Below the working is a numerical example of an elliptic equation solved for all three roots,
and a hyperbolic equation solved for its single positive root.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head127" xml:space="preserve" xml:lang="lat">
e.22.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s843" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda.
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s844" xml:space="preserve">
Hyperbolica <lb/>
æquatio
<lb/>[<emph style="it">tr: 
A hyperbolic equation
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f191v" o="191v" n="382"/>
<pb file="add_6783_f192" o="192" n="383"/>
<div xml:id="echoid-div131" type="page_commentary" level="2" n="131">
<p>
<s xml:id="echoid-s845" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s845" xml:space="preserve">
The square term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo><mi>p</mi><mi>p</mi><mi>a</mi><mo>+</mo><mn>3</mn><mi>r</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>=</mo><mn>2</mn><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math> is removed by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>r</mi><mo>-</mo><mi>e</mi></mstyle></math>.
Similarly, the square term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>3</mn><mi>r</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>p</mi><mi>p</mi><mi>a</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math> is removed by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>e</mi><mo>-</mo><mi>r</mi></mstyle></math>. <lb/>
Conjugate pairs A and B, and C and D, are marked by lines from one column to another. <lb/>
Below the working are two numerical examples of elliptic equations solved for all three roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head128" xml:space="preserve" xml:lang="lat">
e.21.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s847" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda.
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f192v" o="192v" n="384"/>
<pb file="add_6783_f193" o="193" n="385"/>
<div xml:id="echoid-div132" type="page_commentary" level="2" n="132">
<p>
<s xml:id="echoid-s848" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s848" xml:space="preserve">
The square term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>3</mn><mi>r</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>p</mi><mi>p</mi><mi>a</mi><mo>=</mo><mn>2</mn><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math> is removed by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>e</mi><mo>-</mo><mi>r</mi></mstyle></math>.
Similarly, the square term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo><mi>p</mi><mi>p</mi><mi>a</mi><mo>-</mo><mn>3</mn><mi>r</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math> is removed by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mo>-</mo><mi>e</mi><mo>-</mo><mi>r</mi></mstyle></math>. <lb/>
Conjugate pairs A and B, and C and D, are marked by lines from one column to another. <lb/>
Below the working are two numerical examples of elliptic equations solved for all three roots,
and one hyperbolic equations solved for its single positive root.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head129" xml:space="preserve" xml:lang="lat">
e.20.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s850" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda.
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s851" xml:space="preserve">
Hyperbolica <lb/>
æquatio.
<lb/>[<emph style="it">tr: 
A hyperbolic equation
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f193v" o="193v" n="386"/>
<pb file="add_6783_f194" o="194" n="387"/>
<div xml:id="echoid-div133" type="page_commentary" level="2" n="133">
<p>
<s xml:id="echoid-s852" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s852" xml:space="preserve">
The square term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mi>p</mi><mi>p</mi><mi>a</mi><mo>+</mo><mn>3</mn><mi>r</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>=</mo><mn>2</mn><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math> is removed by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>r</mi><mo>-</mo><mi>e</mi></mstyle></math>.
Similarly, the square term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>3</mn><mi>r</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>p</mi><mi>p</mi><mi>a</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math> is removed by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>e</mi><mo>+</mo><mi>r</mi></mstyle></math>. <lb/>
Conjugate pairs A and B, and C and D, are marked by lines from one column to another. <lb/>
Below the working are two numerical examples of elliptic equations solved for all three roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head130" xml:space="preserve" xml:lang="lat">
e.19.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s854" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda.
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f194v" o="194v" n="388"/>
<pb file="add_6783_f195" o="195" n="389"/>
<div xml:id="echoid-div134" type="page_commentary" level="2" n="134">
<p>
<s xml:id="echoid-s855" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s855" xml:space="preserve">
The square term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>3</mn><mi>r</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>p</mi><mi>p</mi><mi>a</mi><mo>=</mo><mn>2</mn><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math> is removed by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>e</mi><mo>-</mo><mi>r</mi></mstyle></math>.
Similarly, the square term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>3</mn><mi>r</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>p</mi><mi>p</mi><mi>a</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math> is removed by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mo>-</mo><mi>e</mi><mo>-</mo><mi>r</mi></mstyle></math>. <lb/>
Conjugate pairs A and B, and C and D, are marked by lines from one column to another. <lb/>
Below the working are two numerical examples of elliptic equations solved for all three roots,
and two of hyperbolic equations solved for their single positive roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head131" xml:space="preserve" xml:lang="lat">
e.18.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s857" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda.
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s858" xml:space="preserve">
Hyperbolica <lb/>
æquatio.
<lb/>[<emph style="it">tr: 
A hyperbolic equation
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f195v" o="195v" n="390"/>
<pb file="add_6783_f196" o="196" n="391"/>
<div xml:id="echoid-div135" type="page_commentary" level="2" n="135">
<p>
<s xml:id="echoid-s859" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s859" xml:space="preserve">
The square term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>r</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>=</mo><mn>2</mn><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math> is removed by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>r</mi><mo>-</mo><mi>e</mi></mstyle></math>.
Similarly, the square term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>3</mn><mi>r</mi><mi>a</mi><mi>a</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math> is removed by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>e</mi><mo>-</mo><mi>r</mi></mstyle></math>. <lb/>
In this case, all the resulting equations are elliptic.
Conjugate pairs A and B, and C and D, are marked by lines from one column to another. <lb/>
Below the working are two numerical examples of elliptic equations solved for all three roots. <lb/>
For a rough version of this page see Add MS 6783, f. 65.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head132" xml:space="preserve" xml:lang="lat">
e.17.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s861" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda.
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s862" xml:space="preserve">
fiat antithesis
<lb/>[<emph style="it">tr: 
use the anithesis
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s863" xml:space="preserve">
fiat antithesis
<lb/>[<emph style="it">tr: 
use the antithesis
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s864" xml:space="preserve">
Sub ista differentia. omnes æquationes sunt eliptica <lb/>
nulla autem hyperbolica.
<lb/>[<emph style="it">tr: 
In this case, all equations are elliptic, and none hyperbolic.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f196v" o="196v" n="392"/>
<pb file="add_6783_f197" o="197" n="393"/>
<div xml:id="echoid-div136" type="page_commentary" level="2" n="136">
<p>
<s xml:id="echoid-s865" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s865" xml:space="preserve">
The square term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>r</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>=</mo><mn>2</mn><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math> is removed by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>e</mi><mo>-</mo><mi>r</mi></mstyle></math>. <lb/>
Similarly, the square term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mn>3</mn><mi>r</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math> is removed by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mo>-</mo><mi>e</mi><mo>-</mo><mi>r</mi></mstyle></math>. <lb/>
This leads to the conjugates of the equation obtained by the first method.
Conjugate pairs A and B, and C and D, are connected by lines from one column to the other. <lb/>
Below the working are two numerical examples of elliptic equations solved for all three roots. <lb/>
At the end of the page, Harrot summarizes the relationship
between the positive roots of an equation and its conjugate in the elliptic case.
He also repeats the claim already made on e.15 (Ad MS 6783, f.198) that hyperbolic equations,
with just one (positive) root, do not have conjugates. <lb/>
For a rough version of this page see Add MS 6783, f. 67.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head133" xml:space="preserve" xml:lang="lat">
e.16.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s867" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda.
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s868" xml:space="preserve">
fiat antithesis
<lb/>[<emph style="it">tr: 
use the antithesis
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s869" xml:space="preserve">
fiat antithesis
<lb/>[<emph style="it">tr: 
use the antithesis
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s870" xml:space="preserve">
Nota.
<lb/>[<emph style="it">tr: 
Note
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s871" xml:space="preserve">
Est autem notandum quod æquationes coniugatarum potes, semper sunt <lb/>
elipticæ: et habent radices tam negatas quam affirmatas.
<lb/>[<emph style="it">tr: 
It is moreover to be noted that equations that can have conjugates are always elliptic;
and they have both negative and positive roots.
</emph>]<lb/>
</s>
<s xml:id="echoid-s872" xml:space="preserve">
Negatæ <lb/>
sunt semper, æquationis coniugatæ, affirmatæ: et contra.
<lb/>[<emph style="it">tr: 
The negatives are always, in the conjugate equation, positive; and vice versa.
</emph>]<lb/>
</s>
<s xml:id="echoid-s873" xml:space="preserve">
Quod in <lb/>
chartis (d) de generatione æquationum potuit observari.
<lb/>[<emph style="it">tr: 
Which may be observed in sheets lettered d, on the generatin of equations.
</emph>]<lb/>
[<emph style="it">Note: 
Sheets d are Add MS 6783, f. 183 to f. 163.
 </emph>]<lb/>
</s>
<s xml:id="echoid-s874" xml:space="preserve">
Hyperbolicæ <lb/>
æquationes non sunt coniugatarum potes, ac ideo unam habent so-<lb/>
lummodo radicem.
<lb/>[<emph style="it">tr: 
Hyperbolic equations cannot have conjugates, and therefore have only one root.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s875" xml:space="preserve">
Æquatio hyperbolica quæ non <lb/>
habet coniugatam:
<lb/>[<emph style="it">tr: 
A hyperbolic equation which does not have a conjugate:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s876" xml:space="preserve">
Neque igitur æquatio proposita.
<lb/>[<emph style="it">tr: 
Nor, therefore, does the equation proposed.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f197v" o="197v" n="394"/>
<pb file="add_6783_f198" o="198" n="395"/>
<div xml:id="echoid-div137" type="page_commentary" level="2" n="137">
<p>
<s xml:id="echoid-s877" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s877" xml:space="preserve">
The square term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>x</mi><mi>x</mi><mi>x</mi><mo>=</mo><mo>-</mo><mn>3</mn><mi>r</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> is removed by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>e</mi><mo>+</mo><mi>r</mi></mstyle></math>.
Below the working is numerical example. <lb/>
Similarly, the square term of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo><mn>3</mn><mi>r</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mi>x</mi><mi>x</mi></mstyle></math> is removed by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>r</mi><mo>-</mo><mi>e</mi></mstyle></math>. <lb/>
This leads to the conjugate of the equation obtained by the first method.
However, under the constraints that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math> are positive,
such an equation has no positive roots, as Harriot shows in his 'Demonstratio' on the right.
At this stage, he therefore regards such an equation as 'impossible'.
Later, in sheet f.14 (Add MS 6783, f. 157),
he will recognize, and use, the fact that the conjugate equation has a pair of imaginary roots. <lb/>
For a rough version of this page see Add MS 6783, f. 91.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head134" xml:space="preserve" xml:lang="lat">
e.15.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s879" xml:space="preserve">
Æquatio adventitia <lb/>
reducenda.
<lb/>[<emph style="it">tr: 
The kind of equation to be reduced
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s880" xml:space="preserve">
Hyperbolica <lb/>
æquatio.
<lb/>[<emph style="it">tr: 
A hyperbolic equation
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s881" xml:space="preserve">
Hyperbolica <lb/>
æquatio.
<lb/>[<emph style="it">tr: 
A hyperbolic equation
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s882" xml:space="preserve">
æquatio coniugata illi quæ 1<emph style="it">o</emph> <lb/>
si esset possiblis.
<lb/>[<emph style="it">tr: 
An equation conjugate to that in the first calculation, if that were possible.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s883" xml:space="preserve">
Sed non est possibilis.
<lb/>[<emph style="it">tr: 
But it is not possible.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s884" xml:space="preserve">
Demonstratio.
<lb/>[<emph style="it">tr: 
Demonstration
</emph>]<lb/>
</s>
<lb/>
<ommision/>
<lb/>
<s xml:id="echoid-s885" xml:space="preserve">
absurdum
<lb/>[<emph style="it">tr: 
absurd
</emph>]<lb/>
</s>
<lb/>
<ommision/>
<lb/>
<s xml:id="echoid-s886" xml:space="preserve">
absurdum
<lb/>[<emph style="it">tr: 
absurd
</emph>]<lb/>
</s>
<lb/>
<ommision/>
<lb/>
<s xml:id="echoid-s887" xml:space="preserve">
absurdum, quia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>&lt;</mo><mi>r</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
absurd becasue <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>&lt;</mo><mi>r</mi></mstyle></math>
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s888" xml:space="preserve">
Ergo: æquatio illa coniugata <lb/>
non est possibilis.
<lb/>[<emph style="it">tr: 
Therefore that conjugate equation is not possible.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f198v" o="198v" n="396"/>
<pb file="add_6783_f199" o="199" n="397"/>
<div xml:id="echoid-div138" type="page_commentary" level="2" n="138">
<p>
<s xml:id="echoid-s889" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s889" xml:space="preserve">
This is the first page of a 4-page subsection of Section f,
in which Harriot continues to remove the cube term from a general quartic.
Harriot's colleague Torporley later denoted this subsection by the letter ϛ (see Stedall 2003, 276–283). <lb/>
On this page, Harriot notes that apart from changes of sign, these equations are equivalent to pairs already treated
(see, for example, Add MS 6783, f. 118), which enables him in subsequent pages to treat four equations at a time.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head135" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s891" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s892" xml:space="preserve">
Consectarium.
</s>
<lb/>
<s xml:id="echoid-s893" xml:space="preserve">
<emph style="st">Hic</emph> apparet quod hic omnia sunt eadem cum antecedentis binis <lb/>
coniugatis quæ sunt his contrariæ; nisi quod <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math> contrarium habet signum <lb/>
et ita erit in omnibus contrarijs.
<lb/>[<emph style="it">tr: 
It is clear that here all are the same as the preceding pairs of conjugates which are here reversed;
except that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math> has the opposite sign.
</emph>]<lb/>
</s>
<s xml:id="echoid-s894" xml:space="preserve">
In sequentibus igitur compendium usurpandum.
<lb/>[<emph style="it">tr: 
In what follows it can therefore be put more briefly.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f199v" o="199v" n="398"/>
<pb file="add_6783_f200" o="200" n="399"/>
<div xml:id="echoid-div139" type="page_commentary" level="2" n="139">
<p>
<s xml:id="echoid-s895" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s895" xml:space="preserve">
A continuation of Add MS 6783, f. 200.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head136" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s897" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f200v" o="200v" n="400"/>
<pb file="add_6783_f201" o="201" n="401"/>
<div xml:id="echoid-div140" type="page_commentary" level="2" n="140">
<p>
<s xml:id="echoid-s898" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s898" xml:space="preserve">
A continuation of Add MS 6783, f. 200 and f. 201.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head137" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s900" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f201v" o="201v" n="402"/>
<pb file="add_6783_f202" o="202" n="403"/>
<div xml:id="echoid-div141" type="page_commentary" level="2" n="141">
<p>
<s xml:id="echoid-s901" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s901" xml:space="preserve">
A continuation of Add MS 6783, f. 200, f. 201, and f. 202.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head138" xml:space="preserve" xml:lang="lat">
f. De resolutione æquationum per reductionem.
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s903" xml:space="preserve">
Æquationes adventitiæ <lb/>
reducendæ
<lb/>[<emph style="it">tr: 
The kind of equations to be reduced.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f202v" o="202v" n="404"/>
<pb file="add_6783_f203" o="203" n="405"/>
<pb file="add_6783_f203v" o="203v" n="406"/>
<pb file="add_6783_f204" o="204" n="407"/>
<div xml:id="echoid-div142" type="page_commentary" level="2" n="142">
<p>
<s xml:id="echoid-s904" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s904" xml:space="preserve">
Harriot continues his exploration of the equation generated by 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><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>f</mi><mo maxsize="1">)</mo><mo>=</mo><mn>0</mn></mstyle></math>.
Here he invesitgates the conditions required for two terms, in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>d</mi></mstyle></math> to vanish simultaneously.
This leads him to values of <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> which are clearly imaginary or,
in the language of Harriot's time, 'impossible' (since <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 both supposed positive).
This is a contradiction since <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> was supposed positive.
Harriot does not seem concerned about this, however, and goes on to add and multiply <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> without comment.
When Warner came to edit these pages in the <emph style="it">Artis analyticae praxis</emph> (1631),
he was troubled by these manipulations and omitted them, stating that they were 'not clear'.
The contradiction lingers in the fact that Harriot started from an equation with three positive roots,
but ends up with an equation with only two, even though his procedures have done nothing to change the roots themselves.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head139" xml:space="preserve">
d.7.2<emph style="super">o</emph>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s906" xml:space="preserve">
Tollere, (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>)
</s>
<lb/>
<s xml:id="echoid-s907" xml:space="preserve">
ponendum
</s>
</p>
<p>
<s xml:id="echoid-s908" xml:space="preserve">
Tum coefficentes primi <lb/>
gradus, et datum planoplanum <lb/>
ita reducuntur.
</s>
</p>
<p>
<s xml:id="echoid-s909" xml:space="preserve">
Ergo æquatio binomia <lb/>
erit:
</s>
</p>
<p>
<s xml:id="echoid-s910" xml:space="preserve">
Et per conversionem <lb/>
signorum erit (ut d.10.)
[<emph style="it">Note: 
Sheet d.10 is Add MS 6783, f. 174.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6783_f204v" o="204v" n="408"/>
<pb file="add_6783_f205" o="205" n="409"/>
<pb file="add_6783_f205v" o="205v" n="410"/>
<pb file="add_6783_f206" o="206" n="411"/>
<pb file="add_6783_f206v" o="206v" n="412"/>
<pb file="add_6783_f207" o="207" n="413"/>
<pb file="add_6783_f207v" o="207v" n="414"/>
<pb file="add_6783_f208" o="208" n="415"/>
<pb file="add_6783_f208v" o="208v" n="416"/>
<pb file="add_6783_f209" o="209" n="417"/>
<pb file="add_6783_f209v" o="209v" n="418"/>
<pb file="add_6783_f210" o="210" n="419"/>
<div xml:id="echoid-div143" type="page_commentary" level="2" n="143">
<p>
<s xml:id="echoid-s911" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s911" xml:space="preserve">
Some quartic equations and their roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s913" xml:space="preserve">
recipr.
<lb/>[<emph style="it">tr: 
reciprocal
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s914" xml:space="preserve">
quadr: <lb/>
recipr.
</s>
<lb/>
<s xml:id="echoid-s915" xml:space="preserve">
Impl: <lb/>
recipr.
</s>
</p>
<pb file="add_6783_f210v" o="210v" n="420"/>
<pb file="add_6783_f211" o="211" n="421"/>
<pb file="add_6783_f211v" o="211v" n="422"/>
<pb file="add_6783_f212" o="212" n="423"/>
<pb file="add_6783_f212v" o="212v" n="424"/>
<pb file="add_6783_f213" o="213" n="425"/>
<pb file="add_6783_f213v" o="213v" n="426"/>
<pb file="add_6783_f214" o="214" n="427"/>
<pb file="add_6783_f214v" o="214v" n="428"/>
<pb file="add_6783_f215" o="215" n="429"/>
<div xml:id="echoid-div144" type="page_commentary" level="2" n="144">
<p>
<s xml:id="echoid-s916" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s916" xml:space="preserve">
Following his lists on Add MS 6783, f. 99 and f. 100
of cases of linear, quadratic, cubic, and quartic equations,
Harriot here offers numerical examples of all possible cases,
including those where all terms in the unknown are negative. <lb/>
He also gives the real roots of each equation, whether positive or negative.
His layout shows very clearly how changing the sign of terms of odd degree
leads to changes of sign in the roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head140" xml:space="preserve" xml:lang="lat">
e.3.2<emph style="super">o</emph> Exempla æquationum in numeris (1
<lb/>[<emph style="it">tr: 
Examples of equations in numbers
</emph>]<lb/>
</head>
<pb file="add_6783_f215v" o="215v" n="430"/>
<pb file="add_6783_f216" o="216" n="431"/>
<div xml:id="echoid-div145" type="page_commentary" level="2" n="145">
<p>
<s xml:id="echoid-s918" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s918" xml:space="preserve">
A continuation of the list from Add MS 6783, f. 215,
completing the list of cubics.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head141" xml:space="preserve" xml:lang="lat">
e.3.3<emph style="super">o</emph> Exempla æquationum in numeris (2
<lb/>[<emph style="it">tr: 
Examples of equations in numbers
</emph>]<lb/>
</head>
<pb file="add_6783_f216v" o="216v" n="432"/>
<pb file="add_6783_f217" o="217" n="433"/>
<div xml:id="echoid-div146" type="page_commentary" level="2" n="146">
<p>
<s xml:id="echoid-s920" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s920" xml:space="preserve">
A continuation of the lists from Add MS 6783, f. 215 and f. 216,
beginning the list of quartics.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head142" xml:space="preserve" xml:lang="lat">
e.3.4<emph style="super">o</emph> Exempla æquationum in numeris (3
<lb/>[<emph style="it">tr: 
Examples of equations in numbers
</emph>]<lb/>
</head>
<pb file="add_6783_f217v" o="217v" n="434"/>
<pb file="add_6783_f218" o="218" n="435"/>
<div xml:id="echoid-div147" type="page_commentary" level="2" n="147">
<p>
<s xml:id="echoid-s922" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s922" xml:space="preserve">
A continuation of the lists from Add MS 6783, f. 215, f. 216, and f. 217,
continuing the list of quartics.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head143" xml:space="preserve" xml:lang="lat">
e.3.5<emph style="super">o</emph> Exempla æquationum in numeris (4
<lb/>[<emph style="it">tr: 
Examples of equations in numbers
</emph>]<lb/>
</head>
<pb file="add_6783_f218v" o="218v" n="436"/>
<pb file="add_6783_f219" o="219" n="437"/>
<div xml:id="echoid-div148" type="page_commentary" level="2" n="148">
<p>
<s xml:id="echoid-s924" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s924" xml:space="preserve">
A continuation of the lists from Add MS 6783, f. 215, f. 216, f. 217, and f. 218,
completing the list of quartics.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head144" xml:space="preserve" xml:lang="lat">
e.3.6<emph style="super">o</emph> Exempla æquationum in numeris (5
<lb/>[<emph style="it">tr: 
Examples of equations in numbers
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s926" xml:space="preserve">
Summa omnium: 120
<lb/>[<emph style="it">tr: 
The sum of all: 120.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f219v" o="219v" n="438"/>
<pb file="add_6783_f220" o="220" n="439"/>
<head xml:id="echoid-head145" xml:space="preserve">
A) Exegesis laterum in quadratis aequationibus
<lb/>[<emph style="it">tr: 
Showing the roots in quadratic equations
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s927" xml:space="preserve">
extrema et media proportio <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. maior pars <lb/>[<emph style="it">tr: 
Extreme and mean proportion, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> the greater part.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s928" xml:space="preserve">
Species exegeseos <lb/>
sive <lb/>
Geometricæ <lb/>
sive <lb/>
Arithmeticæ
<lb/>[<emph style="it">tr: 
General demonstration, whether in geometry or arithmetic.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s929" xml:space="preserve">
sed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> maior, quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> <lb/>
Inde ista æquatio falsa.
<lb/>[<emph style="it">tr: 
but <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, hence this equation is false.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s930" xml:space="preserve">
prima æquatio si per conversionem <emph style="super">signorum</emph> <lb/>
reducatur ad istam: solutio fit <lb/>
per residuum respondens.
<lb/>[<emph style="it">tr: 
the first equation by change of signs is reduced to this; the solution comes about by the corresponding residual.
</emph>]<lb/>
<lb/>[<emph style="it">tr: 
An apotome is a quantity of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>a</mi></mrow></msqrt><mo>+</mo><mi>b</mi></mstyle></math>; the corresponding residual takes the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>a</mi></mrow></msqrt><mo>-</mo><mi>b</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s931" xml:space="preserve">
Geometricæ <lb/>
sive <lb/>
Arithmeticæ
<lb/>[<emph style="it">tr: 
In geometry or arithmetic.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s932" xml:space="preserve">
fiat positio ut 3<emph style="super">æ</emph> inferioribus
<lb/>[<emph style="it">tr: 
Let the position be as the third, below.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s933" xml:space="preserve">
Geometricæ <lb/>
sive <lb/>
Arithmeticæ
<lb/>[<emph style="it">tr: 
In geometry or arithmetic.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f220v" o="220v" n="440"/>
<pb file="add_6783_f221" o="221" n="441"/>
<pb file="add_6783_f221v" o="221v" n="442"/>
<pb file="add_6783_f222" o="222" n="443"/>
<pb file="add_6783_f222v" o="222v" n="444"/>
<pb file="add_6783_f223" o="223" n="445"/>
<pb file="add_6783_f223v" o="223v" n="446"/>
<pb file="add_6783_f224" o="224" n="447"/>
<pb file="add_6783_f224v" o="224v" n="448"/>
<pb file="add_6783_f225" o="225" n="449"/>
<pb file="add_6783_f225v" o="225v" n="450"/>
<pb file="add_6783_f226" o="226" n="451"/>
<head xml:id="echoid-head146" xml:space="preserve" xml:lang="lat">
1.) Si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>K</mi></mstyle></math> sit maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>K</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s934" xml:space="preserve">
Puto quod Non
<lb/>[<emph style="it">tr: 
I think not.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f226v" o="226v" n="452"/>
<head xml:id="echoid-head147" xml:space="preserve" xml:lang="lat">
1) 2<emph style="super">o</emph> 3<emph style="super">o</emph>
De exegesi lateris in sequenti æquatione cubica <lb/>
ubi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>K</mi></mstyle></math> maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
On showing the root in the following cubic equations where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>K</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>.
</emph>]<lb/>
</head>
<pb file="add_6783_f227" o="227" n="453"/>
<head xml:id="echoid-head148" xml:space="preserve" xml:lang="lat">
1) 2<emph style="super">o</emph> De exegesi lateris in sequenti æquatione cubica <lb/>
ubi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>K</mi></mstyle></math> maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>. Non.
<lb/>[<emph style="it">tr: 
On showing the root in the following cubic equations where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>K</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>. No.
</emph>]<lb/>
</head>
<pb file="add_6783_f227v" o="227v" n="454"/>
<pb file="add_6783_f228" o="228" n="455"/>
<head xml:id="echoid-head149" xml:space="preserve" xml:lang="lat">
1) De exegesi lateris. Si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>K</mi></mstyle></math> maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
On showing the root; if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>K</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>.
</emph>]<lb/>
</head>
<pb file="add_6783_f228v" o="228v" n="456"/>
<pb file="add_6783_f229" o="229" n="457"/>
<pb file="add_6783_f229v" o="229v" n="458"/>
<pb file="add_6783_f230" o="230" n="459"/>
<p>
<s xml:id="echoid-s935" xml:space="preserve">
rationall examples
</s>
</p>
<p>
<s xml:id="echoid-s936" xml:space="preserve">
Bombellicæ æquationes
<lb/>[<emph style="it">tr: 
Bombelli's equations
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f230v" o="230v" n="460"/>
<pb file="add_6783_f231" o="231" n="461"/>
<head xml:id="echoid-head150" xml:space="preserve" xml:lang="">
Ad rationales æquationes <lb/>
cubicas <lb/>[<emph style="it">tr: 
On rational cubic equations
</emph>]<lb/>
</head>
<pb file="add_6783_f231v" o="231v" n="462"/>
<pb file="add_6783_f232" o="232" n="463"/>
<div xml:id="echoid-div149" type="page_commentary" level="2" n="149">
<p>
<s xml:id="echoid-s937" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s937" xml:space="preserve">
This sheet refers to Stevin's <emph style="it">L'arithmétique ... aussi l'algebre</emph> (1585), page 309.
On page 308, Stevin wrote the following, about equations he described as 'imparfaicte' (imperfect).
These are equations with three real roots,
but where the solution formula appears at first sight to give imaginary roots.
</s>
<lb/>
<quote xml:lang="fra">
Il auient en aucuns exemples de ceste difference, que le quarré de la moitie du (0) donné,
sera moindre que le cube du tiers du nombre de multitude de (1) donnée;
D'ou s'ensuit que le mesme cube, ne se pourra soubstraire d'iceluy quarré,
comme veut la reigle de la precedent construction; de sorte que ceste premiere difference
(ensemble aucuns exemples des problemes suiuans, que se conuertissent en icelle) est encore imparfaicte.
Rafael Bombelle la solve par diction de plus de moins &amp; moins de moins en ceste sorte:
<lb/>[<emph style="it">tr: 
It happens in some examples of this type, that the square of half of the given term will be less
than the cube of a third of the number of the linear term.
From which it follows that the same cube cannot be subtracted from that square,
as required by the rule of the preceding construction; so that this first type
(together with some examples in the following problems, which convert to it) is still imperfect.
Rafael Bombelli solves it by speaking of <emph style="it">more of less</emph> and <emph style="it">less of less</emph>
in this way.
</emph>]<lb/>
</quote>
<lb/>
<s xml:id="echoid-s938" xml:space="preserve">
Stevin went on to give a solution of 1(3) = 30(1) + 36 (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>3</mn><mn>0</mn><mi>x</mi><mo>-</mo><mn>6</mn></mstyle></math>)
involving <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mo>-</mo><mn>6</mn><mn>7</mn><mn>6</mn></mrow></msqrt></mstyle></math>. He also observed that 6 is a solution. <lb/>
Harriot worked 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>3</mn><mn>0</mn><mi>a</mi><mo>=</mo><mn>3</mn><mn>6</mn></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s940" xml:space="preserve">
problema Bombelli non solvitur multiplicatione cubis. Stev. 309.
<lb/>[<emph style="it">tr: 
A problem not solved by Bombelli, by multiplication of the roots. Stevin page 309.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s941" xml:space="preserve">
Duplicatio lateris.
<lb/>[<emph style="it">tr: 
Doubling the root</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s942" xml:space="preserve">
Decuplatio lateris
<lb/>[<emph style="it">tr: 
Multiplying the root by 10.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f232v" o="232v" n="464"/>
<pb file="add_6783_f233" o="233" n="465"/>
<div xml:id="echoid-div150" type="page_commentary" level="2" n="150">
<p>
<s xml:id="echoid-s943" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s943" xml:space="preserve">
This is the first of a set of 10 pages of notes on Viète's
<emph style="it">Effectionum geometricarum canonica recensio</emph> (1593),
which supplies 20 Euclidean constructions of lines in given arithmetical or proportional relationships.
Propositions 1 and 2, for example, simply show how to add or subtract one line to or from another.
Proposition 3 gives the standard Euclidean construction for three lines in geometric proportion
(<emph style="it">Elements VI.13</emph>). <lb/>
On this page, Harriot gives his own versions of Propositions 5 and 6.
</s>
<lb/>
<quote xml:lang="lat">
Propositio V. <lb/>
Datis duabus lineis rectis, invenire mediam inter eas proportionalem.
</quote>
<lb/>
<quote>
Given two straight lines, to find their mean proportional.
</quote>
<lb/>
<quote xml:lang="lat">
Propositio VI. <lb/>
Datis duabus lineis rectis, invenire tertiam proportionalem.
</quote>
<lb/>
<quote>
Given two straight lines, to find a third proportional.
</quote>
<lb/>
<s xml:id="echoid-s944" xml:space="preserve">
Harriot, following Viète, describes the first as the work of multiplication, and the second as the work of division.
This can be seen more clearly from Harriot's algebraic treatment than from Viète's geometric construction.
This, in Proposition 5, the square of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> (the line to be constructed) represents the product of
the given lines <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 Proposition 6, the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> (to be constructed) represents the quotient <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> of
the given quantites <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</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>
<head xml:id="echoid-head151" xml:space="preserve" xml:lang="lat">
a.) Effectiones geometricæ
<lb/>[<emph style="it">tr: 
Geometrical constructions
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s946" xml:space="preserve">
Invenire, quadratum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>b</mi><mi>c</mi></mstyle></math> <lb/>
Opus multiplicationis.)
<lb/>[<emph style="it">tr: 
To find a square equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> (the work of multiplication)
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s947" xml:space="preserve">
fiat periferia cuius diameter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> <lb/>
<lb/>[...]<lb/> <lb/>
fit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>d</mi></mstyle></math> perpendicularis, signata, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
Construct the circumference whose diameter is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>. <lb/>
<lb/>[...]<lb/> <lb/>
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>d</mi></mstyle></math> be the perpendicular, denoted <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-s948" xml:space="preserve">
Aliter <lb/>
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>=</mo><mi>b</mi><mi>c</mi></mstyle></math> <lb/>
et quæritur. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> <lb/>
ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>,</mo><mi>a</mi><mo>:</mo><mi>a</mi><mo>,</mo><mi>c</mi></mstyle></math>. <lb/>
Igitur <lb/>
Inter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> inveniatur <lb/>
media proportionalis, <lb/>
et signetur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
Another way. <lb/>
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>=</mo><mi>b</mi><mi>c</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> is sought, that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>:</mo><mi>a</mi><mo>=</mo><mi>a</mi><mo>:</mo><mi>c</mi></mstyle></math>. <lb/>
Therefore between <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> there may be found a mean proportional, denoted by <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-s949" xml:space="preserve">
Invenire, planum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math> <lb/>
cuius unum <lb/>
latus datum, nempe, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. <lb/>
Opus applicationis.) <lb/>
fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi><mo>=</mo><mi>b</mi></mstyle></math>. <lb/>
et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>f</mi><mo>=</mo><mi>d</mi></mstyle></math>. perpendicularis. <lb/>
agatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi></mstyle></math>. et secetur bisariam in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>. <lb/>
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>a</mi></mstyle></math>, perpendicularis, quæ secabit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> lineam <emph style="super">productam</emph> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/>
centro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. intervallo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, fiat periferia. <lb/>
quæ secabit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> productam in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> puncto. <lb/>
notetur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>c</mi></mstyle></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
To find a plane equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi></mstyle></math>, of which one side is given, namely <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> (the work of division). <lb/>
Construct <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi><mo>=</mo><mi>b</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>f</mi><mo>=</mo><mi>d</mi></mstyle></math>, the perpendicular.
Join <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi></mstyle></math> and let it be bisected at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>.
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>a</mi></mstyle></math> be a perpendicular, which cuts the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> produced at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>.
With centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> and radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mn>4</mn><mi>c</mi><mi>o</mi><mi>n</mi><mi>s</mi><mi>t</mi><mi>r</mi><mi>u</mi><mi>c</mi><mi>t</mi><mi>t</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>r</mi><mi>c</mi><mi>u</mi><mi>m</mi><mi>f</mi><mi>e</mi><mi>r</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mo>,</mo><mi>w</mi><mi>h</mi><mi>i</mi><mi>c</mi><mi>h</mi><mi>c</mi><mi>u</mi><mi>t</mi><mi>s</mi></mstyle></math> bf <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>r</mi><mi>o</mi><mi>d</mi><mi>u</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>t</mi><mi>h</mi><mi>e</mi><mi>p</mi><mi>o</mi><mi>i</mi><mi>n</mi><mi>t</mi></mstyle></math> c <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>.</mo></mstyle></math><lb/>
It may be observed that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>c</mi></mstyle></math> is <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-s950" xml:space="preserve">
Aliter <lb/>
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math> <lb/>
et quæritur. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> <lb/>
ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>,</mo><mi>d</mi><mo>:</mo><mi>d</mi><mo>,</mo><mi>e</mi></mstyle></math>. <lb/>
Igitur inveniatur tertia proportio-<lb/>
nalis et signetur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
Another way. <lb/>
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> is sought, that is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>:</mo><mi>d</mi><mo>=</mo><mi>d</mi><mo>:</mo><mi>e</mi></mstyle></math>. <lb/>
Therefore there may be found a third proportional, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f233v" o="233v" n="466"/>
<pb file="add_6783_f234" o="234" n="467"/>
<div xml:id="echoid-div151" type="page_commentary" level="2" n="151">
<p>
<s xml:id="echoid-s951" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s951" xml:space="preserve">
On this page, Harriot works on Propositions 7 and 8 from Viète's
<emph style="it">Effectionum geometricarum canonica recensio</emph> (1593).
</s>
<lb/>
<quote xml:lang="lat">
Propositio VII. <lb/>
Datis trianguli rectanguli duobus lateribus circa rectum, invenire latus tertium.
</quote>
<lb/>
<quote>
Given the two sides of a right-angled triangle around the right angle, to find the third side.
</quote>
<lb/>
<quote xml:lang="lat">
Propositio VIII. <lb/>
Dato latere subtendere angulum rectum trianguli, &amp; uno e reliquis, invenire latus rectum.
</quote>
<lb/>
<quote>
Given the side subtending the right angle of a triangle and one of the others, find the other side.
</quote>
<lb/>
<s xml:id="echoid-s952" xml:space="preserve">
In the upper half of the page, Harriot proves that the triangle with sides <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> is right-angled.
In the lower half of the page he gives a brief algebraic summary of Propositions VII and VIII.
Following Viète, he describes the first as addition (of planes) and the second as subtraction (of planes).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head152" xml:space="preserve" xml:lang="lat">
b.) Effectiones geometricæ
<lb/>[<emph style="it">tr: 
Geometrical constructions
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s954" xml:space="preserve">
Ex analyticis principijs <lb/>
probare quod:
<lb/>[...]<lb/>
Ergo: <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>+</mo><mi>d</mi><mi>d</mi></mstyle></math>. per Antithesin
<lb/>[<emph style="it">tr: 
From analytical principles, to prove that:
<lb/>[...]<lb/>
Therefore, by antithesis, <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>+</mo><mi>d</mi><mi>d</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s955" xml:space="preserve">
Addere <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>f</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>k</mi></mstyle></math> <lb/>
<lb/>[...]<lb/> <lb/>
Tum fiant: <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>d</mi></mstyle></math>, latera circa rectum; ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>d</mi></mstyle></math>.
et ducatur ad signata, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
To add <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>f</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>k</mi></mstyle></math> <lb/>
<lb/>[...]<lb/> <lb/>
Then <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> are the sides around the right angle; thus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>d</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s956" xml:space="preserve">
Addere <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>f</mi><mo>+</mo><mi>g</mi><mi>k</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
To add <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>f</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>k</mi></mstyle></math>
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s957" xml:space="preserve">
Invenire, quadratum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>b</mi><mi>b</mi><mo>-</mo><mi>c</mi><mi>c</mi></mstyle></math> <lb/>
<lb/>[...]<lb/> <lb/>
fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>d</mi></mstyle></math> perpendicularis, notata, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
To find a square equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>-</mo><mi>c</mi><mi>c</mi></mstyle></math> <lb/>
<lb/>[...]<lb/> <lb/>
construct the perpendicular <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>d</mi></mstyle></math>, denoted by <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-s958" xml:space="preserve">
Subducere <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>f</mi><mo>-</mo><mi>g</mi><mi>k</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
To subtract <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>k</mi></mstyle></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>f</mi></mstyle></math>
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f234v" o="234v" n="468"/>
<pb file="add_6783_f235" o="235" n="469"/>
<head xml:id="echoid-head153" xml:space="preserve" xml:lang="lat">
c.) Aequationes canonica omnes binomia <lb/>
quam ad locum planum pertinent.
<lb/>[<emph style="it">tr: 
Canonical equations, all binomial, which pertain to the plane.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s959" xml:space="preserve">
Tres primariæ <lb/>
sub forma canonica.
<lb/>[<emph style="it">tr: 
The three primary equations in canonical form.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s960" xml:space="preserve">
Sub forma irregulari <lb/>
sunt: <lb/>
<lb/>[...]<lb/> <lb/>
quæ debent <lb/>
reduci ad formam <lb/>
canonicam.
<lb/>[<emph style="it">tr: 
Those of irregular form are: <lb/>
<lb/>[...]<lb/>
which must be reduced to canonical form.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s961" xml:space="preserve">
Tres secundariæ <lb/>
sub forma canonica. <lb/>
<lb/>[...]<lb/>
Irregulares ad <lb/>
canonicas debent <lb/>
reduci.
<lb/>[<emph style="it">tr: 
Three secondary equations in canonical form. <lb/>
<lb/>[...]<lb/>
The irregular forms must be reduced to canonical forms.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s962" xml:space="preserve">
Sunt alij ordines <lb/>
secundarij minus in usu.
<lb/>[<emph style="it">tr: 
There are other secondary orders, less in use.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f235v" o="235v" n="470"/>
<pb file="add_6783_f236" o="236" n="471"/>
<div xml:id="echoid-div152" type="page_commentary" level="2" n="152">
<p>
<s xml:id="echoid-s963" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s963" xml:space="preserve">
On this page, Harriot works on Propositions 9 and 10 from Viète's
<emph style="it">Effectionum geometricarum canonica recensio</emph> (1593).
</s>
<lb/>
<quote xml:lang="lat">
Propositio IX. <lb/>
Si fuerint tres lineæ rectæ proportionales:
quadratum minoris extremae adjunctum rectangulo sub differentia extremarum &amp; ipsa minore extrema,
æquatur mediæ quadrato.
</quote>
<lb/>
<quote>
If there are three proportional straight lines,
the square of the lesser extreme added to the product of the difference of the extremes and that lesser extreme,
is equal to the square of the mean.
</quote>
<lb/>
<quote xml:lang="lat">
Propositio X. <lb/>
Si fuerint tres lineæ rectæ proportionales:
quadratum majoris extremae multatum rectangulo sub differentia extremarum &amp; ipsa majore extrema,
æquatur mediæ quadrato.
</quote>
<lb/>
<quote>
If there are three proportional straight lines,
the square of the greater extreme reduced by the product of the difference of the extremes and that greater extreme,
is equal to the square of the mean.
</quote>
<lb/>
<s xml:id="echoid-s964" xml:space="preserve">
In both of these propositions, Viète showed how the standard construction for three proportionals
can lead to the given equation.
Harriot works the other way round: beginning from an equation,
he gives a construction that represents the same relationship geometrically.
This is what he means by 'effectio æquationis' or 'the construction of an equation'.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head154" xml:space="preserve" xml:lang="lat">
d.)	Effectiones geometricae
<lb/>[<emph style="it">tr: 
Geometric constructions
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s966" xml:space="preserve">
Effectio æquationis <lb/>
1.) <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math> vel: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mo>,</mo><mi>c</mi><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
Construction of an equation <lb/>
1.) <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s967" xml:space="preserve">
fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>F</mi><mo>=</mo><mi>b</mi></mstyle></math> <lb/>
secetur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>F</mi></mstyle></math> bisariam in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>. <lb/>
et signentur partes <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>A</mi><mi>F</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. <lb/>
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>D</mi></mstyle></math> perpend: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>d</mi></mstyle></math> <lb/>
Agatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>D</mi></mstyle></math>. <lb/>
centro, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>. intervallo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>D</mi></mstyle></math>. <lb/>
describatur periferia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>D</mi><mi>C</mi></mstyle></math> <lb/>
quæ secabit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>F</mi></mstyle></math> productam <lb/>
in punctes. <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/>
Dico quod <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>C</mi><mo>=</mo><mi>a</mi></mstyle></math>. <lb/>
cui æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>B</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
Construct <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>F</mi><mo>=</mo><mi>b</mi></mstyle></math>, and let it be bisected at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>, and denote the parts <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>A</mi><mi>F</mi></mstyle></math>, by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
Let the perpendicular <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>D</mi></mstyle></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. Connect <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>D</mi></mstyle></math>. With centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> and radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>D</mi></mstyle></math>,
describe the circumference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>D</mi><mi>C</mi></mstyle></math>, which will cut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>F</mi></mstyle></math> produced in the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>. <lb/>
I say that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>C</mi><mo>=</mo><mi>a</mi></mstyle></math>, which is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>B</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s968" xml:space="preserve">
vel binis caracteribus.
<lb/>[<emph style="it">tr: 
or in double letters
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s969" xml:space="preserve">
Effectio æquationis <lb/>
2.) <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>-</mo><mi>b</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math> vel: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>-</mo><mn>2</mn><mi>c</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
Construction of an equation <lb/>
2.) <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>-</mo><mi>b</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>-</mo><mn>2</mn><mi>c</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s970" xml:space="preserve">
constructio ommnino ut supra <lb/>
<lb/>[...]<lb/> <lb/>
vel binis caracteribus.
<lb/>[<emph style="it">tr: 
The construction of everything is as above <lb/>
<lb/>[...]<lb/> <lb/>
or in double letters.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f236v" o="236v" n="472"/>
<pb file="add_6783_f237" o="237" n="473"/>
<div xml:id="echoid-div153" type="page_commentary" level="2" n="153">
<p>
<s xml:id="echoid-s971" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s971" xml:space="preserve">
On this page, Harriot works on Proposition 11 from Viète's
<emph style="it">Effectionum geometricarum canonica recensio</emph> (1593).
</s>
<lb/>
<quote xml:lang="lat">
Propositio XI. <lb/>
Si fuerint tres lineæ rectæ proportionales:
rectangulum sub composita ex extremis &amp; harum altera majore minorve, multatum ejusdem alterius quadrato,
æquatur mediae quadrato.
</quote>
<lb/>
<quote>
If there are three proportional straight lines,
the product of the sum of the extremes and either the greater or the lesser, reduced by the square of the same,
is equal to the square of the mean.
</quote>
<lb/>
<s xml:id="echoid-s972" xml:space="preserve">
In this proposition, as in propositions IX and X,
Viète showed how the standard construction for three proportionals can lead to the given equation.
As on the previous page (Add MS 6783, f. 236), Harriot works the other way round: beginning from the equation,
he gives a construction that represents the same relationship geometrically.
He notes that the construction gives both the positive roots that are to be expected for this case.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head155" xml:space="preserve" xml:lang="lat">
e.)	Effectiones geometricae
<lb/>[<emph style="it">tr: 
Geometric constructions
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s974" xml:space="preserve">
Effectio æquationis <lb/>
3.) <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math> vel: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mo>,</mo><mi>c</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
Construction of an equation <lb/>
3.) <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s975" xml:space="preserve">
fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi><mo>=</mo><mi>b</mi></mstyle></math> <lb/>
secetur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math> bisariam in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>. <lb/>
et signentur partes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. <lb/>
centro, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>. intervallo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>. <lb/>
describatur <emph style="st">circulas</emph> periferia <lb/>
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math> perpendicularis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math> <lb/>
fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>I</mi><mo>=</mo><mi>d</mi></mstyle></math>. <lb/>
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi><mi>D</mi></mstyle></math> parallela <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math>. <lb/>
et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>F</mi></mstyle></math> parallela <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi><mi>A</mi></mstyle></math>. <lb/>
Ergo. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>F</mi><mo>=</mo><mi>I</mi><mi>A</mi><mo>=</mo><mi>d</mi></mstyle></math>. <lb/>
Dico quod <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> erit duplex, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>F</mi></mstyle></math> <emph style="super">vel</emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>C</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
Construct <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi><mo>=</mo><mi>b</mi></mstyle></math>, and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math> be bisected at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>, and denote the parts <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math>, by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
With centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> and radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>, describe a circumference. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math> be perpendicula to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math>.
Construct <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>I</mi><mo>=</mo><mi>d</mi></mstyle></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi><mi>D</mi></mstyle></math> be parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>F</mi></mstyle></math> be parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi><mi>A</mi></mstyle></math>.
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>F</mi><mo>=</mo><mi>I</mi><mi>A</mi><mo>=</mo><mi>d</mi></mstyle></math>. <lb/>
I say that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> will be twofold, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>F</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>C</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f237v" o="237v" n="474"/>
<pb file="add_6783_f238" o="238" n="475"/>
<pb file="add_6783_f238v" o="238v" n="476"/>
<pb file="add_6783_f239" o="239" n="477"/>
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<pb file="add_6783_f242" o="242" n="483"/>
<pb file="add_6783_f242v" o="242v" n="484"/>
<pb file="add_6783_f243" o="243" n="485"/>
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<pb file="add_6783_f244v" o="244v" n="488"/>
<pb file="add_6783_f245" o="245" n="489"/>
<pb file="add_6783_f245v" o="245v" n="490"/>
<pb file="add_6783_f246" o="246" n="491"/>
<pb file="add_6783_f246v" o="246v" n="492"/>
<pb file="add_6783_f247" o="247" n="493"/>
<pb file="add_6783_f247v" o="247v" n="494"/>
<pb file="add_6783_f248" o="248" n="495"/>
<pb file="add_6783_f248v" o="248v" n="496"/>
<pb file="add_6783_f249" o="249" n="497"/>
<pb file="add_6783_f249v" o="249v" n="498"/>
<pb file="add_6783_f250" o="250" n="499"/>
<pb file="add_6783_f250v" o="250v" n="500"/>
<pb file="add_6783_f251" o="251" n="501"/>
<pb file="add_6783_f251v" o="251v" n="502"/>
<pb file="add_6783_f252" o="252" n="503"/>
<pb file="add_6783_f252v" o="252v" n="504"/>
<pb file="add_6783_f253" o="253" n="505"/>
<pb file="add_6783_f253v" o="253v" n="506"/>
<pb file="add_6783_f254" o="254" n="507"/>
<pb file="add_6783_f254v" o="254v" n="508"/>
<pb file="add_6783_f255" o="255" n="509"/>
<pb file="add_6783_f255v" o="255v" n="510"/>
<pb file="add_6783_f256" o="256" n="511"/>
<pb file="add_6783_f256v" o="256v" n="512"/>
<pb file="add_6783_f257" o="257" n="513"/>
<pb file="add_6783_f257v" o="257v" n="514"/>
<pb file="add_6783_f258" o="258" n="515"/>
<head xml:id="echoid-head156" xml:space="preserve">
f.8
</head>
<pb file="add_6783_f258v" o="258v" n="516"/>
<head xml:id="echoid-head157" xml:space="preserve">
f.8
</head>
<pb file="add_6783_f259" o="259" n="517"/>
<pb file="add_6783_f259v" o="259v" n="518"/>
<pb file="add_6783_f260" o="260" n="519"/>
<head xml:id="echoid-head158" xml:space="preserve">
f.8
</head>
<pb file="add_6783_f260v" o="260v" n="520"/>
<pb file="add_6783_f261" o="261" n="521"/>
<pb file="add_6783_f261v" o="261v" n="522"/>
<pb file="add_6783_f262" o="262" n="523"/>
<pb file="add_6783_f262v" o="262v" n="524"/>
<pb file="add_6783_f263" o="263" n="525"/>
<pb file="add_6783_f263v" o="263v" n="526"/>
<pb file="add_6783_f264" o="264" n="527"/>
<pb file="add_6783_f264v" o="264v" n="528"/>
<pb file="add_6783_f265" o="265" n="529"/>
<pb file="add_6783_f265v" o="265v" n="530"/>
<pb file="add_6783_f266" o="266" n="531"/>
<head xml:id="echoid-head159" xml:space="preserve">
f.8
</head>
<pb file="add_6783_f266v" o="266v" n="532"/>
<pb file="add_6783_f267" o="267" n="533"/>
<pb file="add_6783_f267v" o="267v" n="534"/>
<pb file="add_6783_f268" o="268" n="535"/>
<head xml:id="echoid-head160" xml:space="preserve">
f.8
</head>
<pb file="add_6783_f268v" o="268v" n="536"/>
<pb file="add_6783_f269" o="269" n="537"/>
<pb file="add_6783_f269v" o="269v" n="538"/>
<pb file="add_6783_f270" o="270" n="539"/>
<pb file="add_6783_f270v" o="270v" n="540"/>
<head xml:id="echoid-head161" xml:space="preserve">
f.8
</head>
<pb file="add_6783_f271" o="271" n="541"/>
<pb file="add_6783_f271v" o="271v" n="542"/>
<pb file="add_6783_f272" o="272" n="543"/>
<head xml:id="echoid-head162" xml:space="preserve">
D.
</head>
<pb file="add_6783_f272v" o="272v" n="544"/>
<pb file="add_6783_f273" o="273" n="545"/>
<div xml:id="echoid-div154" type="page_commentary" level="2" n="154">
<p>
<s xml:id="echoid-s976" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s976" xml:space="preserve">
The removal of the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> from the equation on Add MS 6783, f. 274. <lb/>
See also Add MS 6783, f. 177 (d.7).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head163" xml:space="preserve">
at<emph style="super">2</emph>)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s978" xml:space="preserve">
ad tollenda <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
for the removal of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f273v" o="273v" n="546"/>
<pb file="add_6783_f274" o="274" n="547"/>
<div xml:id="echoid-div155" type="page_commentary" level="2" n="155">
<p>
<s xml:id="echoid-s979" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s979" xml:space="preserve">
See also Add MS 6783, f. 177, where the same equation is derived,
but there the possibility of the negative root <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mo>-</mo><mi>f</mi></mstyle></math> is not taken into account.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head164" xml:space="preserve">
at)
</head>
<pb file="add_6783_f274v" o="274v" n="548"/>
<pb file="add_6783_f275" o="275" n="549"/>
<div xml:id="echoid-div156" type="page_commentary" level="2" n="156">
<p>
<s xml:id="echoid-s981" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s981" xml:space="preserve">
The removal of the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> from the equation on Add MS 6783, f. 274. <lb/>
See also Add MS 6783, f. 177 (d.7).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head165" xml:space="preserve">
at<emph style="super">3</emph>)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s983" xml:space="preserve">
ad tollenda <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
for the removal of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f275v" o="275v" n="550"/>
<pb file="add_6783_f276" o="276" n="551"/>
<pb file="add_6783_f276v" o="276v" n="552"/>
<pb file="add_6783_f277" o="277" n="553"/>
<pb file="add_6783_f277v" o="277v" n="554"/>
<pb file="add_6783_f278" o="278" n="555"/>
<pb file="add_6783_f278v" o="278v" n="556"/>
<pb file="add_6783_f279" o="279" n="557"/>
<pb file="add_6783_f279v" o="279v" n="558"/>
<pb file="add_6783_f280" o="280" n="559"/>
<pb file="add_6783_f280v" o="280v" n="560"/>
<pb file="add_6783_f281" o="281" n="561"/>
<div xml:id="echoid-div157" type="page_commentary" level="2" n="157">
<p>
<s xml:id="echoid-s984" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s984" xml:space="preserve">
See Add MS 6784, f. 415.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6783_f281v" o="281v" n="562"/>
<pb file="add_6783_f282" o="282" n="563"/>
<div xml:id="echoid-div158" type="page_commentary" level="2" n="158">
<p>
<s xml:id="echoid-s986" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s986" xml:space="preserve">
Removal of the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> from the product <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><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>f</mi><mo maxsize="1">)</mo></mstyle></math>. <lb/>
See also Add MS 6783, f. 175 (d.9).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head166" xml:space="preserve">
as)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s988" xml:space="preserve">
pro charta ag)
<lb/>[<emph style="it">tr: 
for sheet <emph style="it">ag</emph>
</emph>]<lb/>
[<emph style="it">Note: 
Sheet ag is Add MS 6783, f. 297.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s989" xml:space="preserve">
ad expurgandum. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
for the removal of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s990" xml:space="preserve">
expurgat datum <lb/>
ut in dorso. ar)
<lb/>[<emph style="it">tr: 
the given term disappears, as on the back of <emph style="it">ar</emph>
</emph>]<lb/>
[<emph style="it">Note: 
The back of sheet ar is Add MS 6783, f. 285v.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s991" xml:space="preserve">
quære alia tempore
<lb/>[<emph style="it">tr: 
as at another time
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s992" xml:space="preserve">
in dorso. ar)
<lb/>[<emph style="it">tr: 
on the back of <emph style="it">ar</emph>
</emph>]<lb/>
[<emph style="it">Note: 
The back of sheet ar is Add MS 6783, f. 285v.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s993" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> sunt affirmata
<lb/>[<emph style="it">tr: 
the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> are positive
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s994" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> sunt negata
<lb/>[<emph style="it">tr: 
the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> are negative
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s995" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> sunt hic semper affir-<lb/>
mata
<lb/>[<emph style="it">tr: 
the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> are here always positive
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s996" xml:space="preserve">
vide supra
<lb/>[<emph style="it">tr: 
see above
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s997" xml:space="preserve">
æquatio est transponenda. et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> fit affirmata. <lb/>
et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> erit negatum: quia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi><mo>&gt;</mo><mi>b</mi><mi>d</mi><mo>+</mo><mi>c</mi><mi>d</mi></mstyle></math> <lb/>
et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> erit affirmaum. quia:
<lb/>[<emph style="it">tr: 
the equation is transposed, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> becomes positive,
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> will be negative because <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi><mo>&gt;</mo><mi>b</mi><mi>d</mi><mo>+</mo><mi>c</mi><mi>d</mi></mstyle></math>,
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> will be positive, because:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s998" xml:space="preserve">
hæc alibi <lb/>
aliter
<lb/>[<emph style="it">tr: 
these elsewhere otherwise
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s999" xml:space="preserve">
sed hic melius
<lb/>[<emph style="it">tr: 
but better here
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f282v" o="282v" n="564"/>
<pb file="add_6783_f283" o="283" n="565"/>
<div xml:id="echoid-div159" type="page_commentary" level="2" n="159">
<p>
<s xml:id="echoid-s1000" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1000" xml:space="preserve">
Further work on the equation in Add MS 6783, f 284.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head167" xml:space="preserve">
ar<emph style="super">3</emph>)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1002" xml:space="preserve">
In primo exemplo: si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>=</mo><mi>b</mi></mstyle></math>, vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. datum erit negativum <lb/>
et tum per metathesin vel conversionem signorum fit æquatio sequentes.
<lb/>[<emph style="it">tr: 
In the first example, if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>=</mo><mi>b</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>=</mo><mi>c</mi></mstyle></math>, the given term is negative,
and then by metathesis or changing the signs, there comes about the following equation.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1003" xml:space="preserve">
nota <lb/>
diligentibus
<lb/>[<emph style="it">tr: 
note to the attentive
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1004" xml:space="preserve">
Si: <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> in primo exemplo.
<lb/>[<emph style="it">tr: 
If <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> in the first example.
</emph>]<lb/>
</s>
<s xml:id="echoid-s1005" xml:space="preserve">
In dorso: aq)
<lb/>[<emph style="it">tr: 
On the back of sheet <emph style="it">aq</emph>
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1006" xml:space="preserve">
hic ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, debet esse minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mfrac><mrow><mn>5</mn><mi>b</mi><mi>b</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msqrt><mo>-</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> <lb/>
quabis est 3, ad 2, et omnes minores.
<lb/>[<emph style="it">tr: 
Here the ratio of <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> must be less than <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><msqrt><mrow><mfrac><mrow><mn>5</mn><mi>b</mi><mi>b</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msqrt><mo>-</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>,
which is about 3 to 2, and anything less.
</emph>]<lb/>
[<emph style="it">Note: 
The back of sheet aq is Add MS 6783, f. 287v.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1007" xml:space="preserve">
alias: si æqualis, tollit datum: si maior, facit datum <lb/>
negativum, et tum per metathesin vel signorum conversionem fit æquatio sequens.
<lb/>[<emph style="it">tr: 
Otherwise, if it is equal, the given term is removed; if greater. the given term is negative,
and then by metathesis or changing signs there comes about the following equation.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1008" xml:space="preserve">
hic, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, semper negatur. <lb/>
et, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, semper affirmatur.
<lb/>[<emph style="it">tr: 
here <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> is always negative and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> is always positive.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1009" xml:space="preserve">
In dorso: aq
<lb/>[<emph style="it">tr: 
On the back of sheet <emph style="it">aq</emph>
</emph>]<lb/>
[<emph style="it">Note: 
The back of sheet aq is Add MS 6783, f. 287v.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1010" xml:space="preserve">
hic ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, debet esse maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mfrac><mrow><mn>5</mn><mi>b</mi><mi>b</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msqrt><mo>-</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> <lb/>
quabis est 5, ad 3, et omnes maiores.
<lb/>[<emph style="it">tr: 
Here the ratio of <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> must be greater than <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><msqrt><mrow><mfrac><mrow><mn>5</mn><mi>b</mi><mi>b</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msqrt><mo>-</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>,
which is about 5 to 3, and anything greater.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f283v" o="283v" n="566"/>
<pb file="add_6783_f284" o="284" n="567"/>
<div xml:id="echoid-div160" type="page_commentary" level="2" n="160">
<p>
<s xml:id="echoid-s1011" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1011" xml:space="preserve">
An investigation of the equation arising from setting the product
<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><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>f</mi><mo maxsize="1">)</mo></mstyle></math> equal to zero,
after removal of the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>, for the cases <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>&lt;</mo><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>&gt;</mo><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>. <lb/>
See also Add MS 6783, f. 175 (d.9).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head168" xml:space="preserve">
ar<emph style="super">2</emph>)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1013" xml:space="preserve">
primo. si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>&lt;</mo><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
first, f <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>&lt;</mo><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1014" xml:space="preserve">
secundo. si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>&gt;</mo><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
second, if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>&gt;</mo><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1015" xml:space="preserve">
In primo æquatione: so <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>, sit minor, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> unitate <lb/>
et in secunda: si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, sit maior, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>, unitate: <lb/>
facies æquationes in numeris integris.
<lb/>[<emph style="it">tr: 
In the first equation, if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> is greater and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is one,
and in the second, if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is greater and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> is one,
the equations may be solved in whole numbers.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1016" xml:space="preserve">
expurgat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> disappears.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f284v" o="284v" n="568"/>
<pb file="add_6783_f285" o="285" n="569"/>
<div xml:id="echoid-div161" type="page_commentary" level="2" n="161">
<p>
<s xml:id="echoid-s1017" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1017" xml:space="preserve">
Removal of the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> from the product <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><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>f</mi><mo maxsize="1">)</mo></mstyle></math>. <lb/>
See also Add MS 6783, f. 175 (d.9).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head169" xml:space="preserve">
ar
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1019" xml:space="preserve">
ad expurgandum. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
for the removal of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/><s xml:id="echoid-s1020" xml:space="preserve">
pro charta ag)
<lb/>[<emph style="it">tr: 
for sheet <emph style="it">ag</emph>
</emph>]<lb/>
[<emph style="it">Note: 
Sheet ag is Add MS 6783, f. 297.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6783_f285v" o="285v" n="570"/>
<p xml:lang="lat">
<s xml:id="echoid-s1021" xml:space="preserve">
pro charta as)
<lb/>[<emph style="it">tr: 
for sheet <emph style="it">as</emph>
</emph>]<lb/>
[<emph style="it">Note: 
Sheet as is Add MS 6783, f. 282.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6783_f286" o="286" n="571"/>
<div xml:id="echoid-div162" type="page_commentary" level="2" n="162">
<p>
<s xml:id="echoid-s1022" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1022" xml:space="preserve">
Further calculations on the equation in Add MS 6783, f. 287, in the cases where <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> or <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>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head170" xml:space="preserve">
aq<emph style="super">2</emph>)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1024" xml:space="preserve">
pro charta aq
<lb/>[<emph style="it">tr: 
for sheet <emph style="it">aq</emph>
</emph>]<lb/>
[<emph style="it">Note: 
Sheet aq is Add MS 6783, f. 287.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1025" xml:space="preserve">
si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> sit maior ratio quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mfrac><mrow><mn>5</mn><mi>b</mi><mi>b</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msqrt><mo>-</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>:
quabis 5 ad 3, et maiores. <lb/>
(in dorso ag)
<lb/>[<emph style="it">tr: 
If <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> is in a greater ratio than <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><msqrt><mrow><mfrac><mrow><mn>5</mn><mi>b</mi><mi>b</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msqrt><mo>-</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>:
about 5 to 3, and more. <lb/>
(on the back of sheet <emph style="it">ag</emph>)
</emph>]<lb/>
[<emph style="it">Note: 
The back of sheet ag is Add MS 6783, f. 297v.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1026" xml:space="preserve">
si minor ratio, quabis 3 ad 2 et minores.
<lb/>[<emph style="it">tr: 
If a smaller ratio, about 3 to 2 and less.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1027" xml:space="preserve">
si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, negatur; tamen <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> potest <lb/>
vel negativi vel affirmavi.
<lb/>[<emph style="it">tr: 
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> is negative; however <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> may be either negative or positive.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1028" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>, et datum, semper affirmantur.
<lb/>[<emph style="it">tr: 
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> and the given term are always positive.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1029" xml:space="preserve">
Verte paginam.
<lb/>[<emph style="it">tr: 
Turn the page.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f286v" o="286v" n="572"/>
<pb file="add_6783_f287" o="287" n="573"/>
<div xml:id="echoid-div163" type="page_commentary" level="2" n="163">
<p>
<s xml:id="echoid-s1030" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1030" xml:space="preserve">
The equation arising from setting the product <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><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>f</mi><mo maxsize="1">)</mo></mstyle></math> equal to zero,
with the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> removed. <lb/>
See also Add MS 6783, f. 175 (d.9), f. 174 (d.10).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head171" xml:space="preserve">
aq)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1032" xml:space="preserve">
examind
<lb/>[<emph style="it">tr: 
to be examined
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1033" xml:space="preserve">
Generatio in dorso <lb/>
chartæ. ap)
<lb/>[<emph style="it">tr: 
The generation is on the back of the sheet, <emph style="it">ap</emph>.
</emph>]<lb/>
[<emph style="it">Note: 
The back of sheet ap is Add MS 6783, f. 288v.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1034" xml:space="preserve">
expurgat. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. et nota consequentiam.
<lb/>[<emph style="it">tr: 
This removes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, and note the consequence.
</emph>]<lb/>
[<emph style="it">Note: 
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> here is an error; it should read <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1035" xml:space="preserve">
retinet. et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> debet esse maior si æquatio habet <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <lb/>
si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> tamen ponatur minor <lb/>
nota consequentiam:
<lb/>[<emph style="it">tr: 
This retains <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is neccessarily greater if the equation has <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>.
If, however, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is supposed smaller, note the consequence.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1036" xml:space="preserve">
quid aq<emph style="it">2</emph>
<lb/>[<emph style="it">tr: 
which is in <emph style="it">aq2</emph>
</emph>]<lb/>
[<emph style="it">Note: 
Sheet aq2 is Add MS 6783, f. 286.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1037" xml:space="preserve">
dorso aq<emph style="it">2</emph>
<lb/>[<emph style="it">tr: 
the back of <emph style="it">aq2</emph>
</emph>]<lb/>
[<emph style="it">Note: 
The back of aq2 is Add MS 6783, f. 286v.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1038" xml:space="preserve">
erit æquatio parabolica.
<lb/>[<emph style="it">tr: 
it will be a parabolic equation
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1039" xml:space="preserve">
sed superioris æquationis <lb/>
propria parabolica est cum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mi>b</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
but the above equation is properly parabolic when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mi>b</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f287v" o="287v" n="574"/>
<div xml:id="echoid-div164" type="page_commentary" level="2" n="164">
<p>
<s xml:id="echoid-s1040" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1040" xml:space="preserve">
Further work on the equation in Add MS 6783, f 284.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s1042" xml:space="preserve">
pro charta ar<emph style="it">2</emph>)
<lb/>[<emph style="it">tr: 
for sheet <emph style="it">ar2</emph>
</emph>]<lb/>
[<emph style="it">Note: 
Sheet ar2 is Add MS 6783, f. 284.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6783_f288" o="288" n="575"/>
<div xml:id="echoid-div165" type="page_commentary" level="2" n="165">
<p>
<s xml:id="echoid-s1043" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1043" xml:space="preserve">
Investigation of the condition <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>c</mi><mo>=</mo><mi>c</mi><mi>d</mi><mo>+</mo><mn>2</mn><mi>b</mi><mi>d</mi></mstyle></math>,
required for the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> to vanish from the product on the previous page, Add MS 6783, f. 289.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head172" xml:space="preserve">
ap)
</head>
<pb file="add_6783_f288v" o="288v" n="576"/>
<div xml:id="echoid-div166" type="page_commentary" level="2" n="166">
<p>
<s xml:id="echoid-s1045" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1045" xml:space="preserve">
Removal of the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> from the product <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><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>f</mi><mo maxsize="1">)</mo></mstyle></math>. <lb/>
See also Add MS 6783, f. 174 (d.10).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s1047" xml:space="preserve">
pro charta aq)
<lb/>[<emph style="it">tr: 
for sheet <emph style="it">aq</emph>
</emph>]<lb/>
[<emph style="it">Note: 
Sheet aq is Add MS 6783, f. 287.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6783_f289" o="289" n="577"/>
<div xml:id="echoid-div167" type="page_commentary" level="2" n="167">
<p>
<s xml:id="echoid-s1048" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1048" xml:space="preserve">
Calculation of the product <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>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>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head173" xml:space="preserve">
ao)
</head>
<pb file="add_6783_f289v" o="289v" n="578"/>
<pb file="add_6783_f290" o="290" n="579"/>
<div xml:id="echoid-div168" type="page_commentary" level="2" n="168">
<p>
<s xml:id="echoid-s1050" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1050" xml:space="preserve">
Removal of the terms in <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>a</mi><mi>a</mi></mstyle></math> from the product <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><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>f</mi><mo maxsize="1">)</mo></mstyle></math>. <lb/>
See also Add MS 6783, f. 173 (d.11).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head174" xml:space="preserve">
an<emph style="super">2</emph>)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1052" xml:space="preserve">
binomium per suum residi
<lb/>[<emph style="it">tr: 
binomials through their residuals
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1053" xml:space="preserve">
Coefficientes sunt continue proportionales <lb/>
tres primæ sunt <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>b</mi></mstyle></math> <lb/>
tres positivæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>c</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
The coefficients are in continued proportion;
the first three are equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, the positive three are equal to <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-s1054" xml:space="preserve">
non convertitur: unde <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi><mo>&gt;</mo><mi>d</mi><mo>+</mo><mi>f</mi></mstyle></math> neccessario
<lb/>[<emph style="it">tr: 
not converted, whence necessarily <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi><mo>&gt;</mo><mi>d</mi><mo>+</mo><mi>f</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f290v" o="290v" n="580"/>
<p xml:lang="lat">
<s xml:id="echoid-s1055" xml:space="preserve">
Nota. insertum in charta aq)
<lb/>[<emph style="it">tr: 
insert for sheet <emph style="it">aq</emph>
</emph>]<lb/>
[<emph style="it">Note: 
Sheet aq is Add MS 6783, f. 287.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6783_f291" o="291" n="581"/>
<div xml:id="echoid-div169" type="page_commentary" level="2" n="169">
<p>
<s xml:id="echoid-s1056" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1056" xml:space="preserve">
Removal of the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> from the product <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><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>f</mi><mo maxsize="1">)</mo></mstyle></math>. <lb/>
See also Add MS 6783, f. 175 (d.9).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head175" xml:space="preserve">
an<emph style="super">1</emph>)
</head>
<pb file="add_6783_f291v" o="291v" n="582"/>
<pb file="add_6783_f292" o="292" n="583"/>
<div xml:id="echoid-div170" type="page_commentary" level="2" n="170">
<p>
<s xml:id="echoid-s1058" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1058" xml:space="preserve">
See also Add MS 6783, f. 173 (d.11) for the derivation and solution of the same quadratic equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head176" xml:space="preserve">
an)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1060" xml:space="preserve">
pro charta. aq) ad tollenda, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
for sheet <emph style="it">aq</emph>, for the removal of the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1061" xml:space="preserve">
vide chartam an1) an2) al)
<lb/>[<emph style="it">tr: 
see sheet <emph style="it">an1</emph>, <emph style="it">an2</emph>, <emph style="it">al</emph>.
</emph>]<lb/>
[<emph style="it">Note: 
Sheets an1, an2, al are Add MS 6783, f. 291, f. 290, f. 294, respectively.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6783_f292v" o="292v" n="584"/>
<pb file="add_6783_f293" o="293" n="585"/>
<div xml:id="echoid-div171" type="page_commentary" level="2" n="171">
<p>
<s xml:id="echoid-s1062" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1062" xml:space="preserve">
See also Add MS 6783, f. 173 (d.11) for another derivation of the same quadratic equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head177" xml:space="preserve">
am)
</head>
<pb file="add_6783_f293v" o="293v" n="586"/>
<pb file="add_6783_f294" o="294" n="587"/>
<div xml:id="echoid-div172" type="page_commentary" level="2" n="172">
<p>
<s xml:id="echoid-s1064" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1064" xml:space="preserve">
For similar working see also Add MS 6783, f. 173 (d.11).
There the same quadratic equation is solved for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>,
but the final quantity under the square root sign is omitted,
and Harriot does not comment (as he does here) on the fact that <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> are imaginary.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head178" xml:space="preserve">
al)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1066" xml:space="preserve">
Hic negativa sunt maiora affirmativis <lb/>
unde hic casus in æquatione fundamentali <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> <lb/>
videtur impossibilis.
<lb/>[<emph style="it">tr: 
Here the negatives are greater than the positives,
whence in this case of the fundamental equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> is seen to be impossible.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1067" xml:space="preserve">
Sheet am is Add MS 6783, f. 293.
Vide igitur sequentem chartam am)
<lb/>[<emph style="it">tr: 
See therefore the following sheet <emph style="it">am</emph>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f294v" o="294v" n="588"/>
<pb file="add_6783_f295" o="295" n="589"/>
<div xml:id="echoid-div173" type="page_commentary" level="2" n="173">
<p>
<s xml:id="echoid-s1068" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1068" xml:space="preserve">
The 4th- and 5th-degree equations in the lower half of the page appear also on
Add MS 6783, f. 170 (d.14), in Harriot's first collection of canonical equations.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head179" xml:space="preserve">
ak)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1070" xml:space="preserve">
Coefficientes sunt contine proportionalis. <lb/>
&amp; sic de omnibus binomijs æquationis <lb/>
sub latere affirmate et postestate negata.
<lb/>[<emph style="it">tr: 
The coefficients are in continued proportion,
and similarly for all other equations with a positive linear term and a negative square.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1071" xml:space="preserve">
alia dispositio:
<lb/>[<emph style="it">tr: 
another arrangement
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f295v" o="295v" n="590"/>
<head xml:id="echoid-head180" xml:space="preserve">
ai)
</head>
<pb file="add_6783_f296" o="296" n="591"/>
<div xml:id="echoid-div174" type="page_commentary" level="2" n="174">
<p>
<s xml:id="echoid-s1072" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1072" xml:space="preserve">
Work on the products <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><mo maxsize="1">(</mo><mi>a</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>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><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>f</mi><mo maxsize="1">)</mo></mstyle></math>. <lb/>
For the former see also Add MS 6783, f. 176 (d.8). <lb/>
For the conditions required for the removal of terms from the latter see
Add MS 6783, f. 174 (d.10), f. 173 (d.11), f. 172 (d.12).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head181" xml:space="preserve">
ah)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1074" xml:space="preserve">
ad chartam ag)
<lb/>[<emph style="it">tr: 
for sheet <emph style="it">ag</emph>
</emph>]<lb/>
[<emph style="it">Note: 
Sheet ag is Add MS 6783, f. 297.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6783_f296v" o="296v" n="592"/>
<div xml:id="echoid-div175" type="page_commentary" level="2" n="175">
<p>
<s xml:id="echoid-s1075" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1075" 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>. <lb/>
See Add MS 6782, f. 323, for an explanation of the terms 'elliptic' and 'Bombellian'. <lb/>
See Add MS 6783, f. 180 (d.4) for the derivation of similar equations.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head182" xml:space="preserve">
af)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1077" xml:space="preserve">
Fundamentum
<lb/>[<emph style="it">tr: 
Foundation
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1078" xml:space="preserve">
Eliptica. <lb/>
seu bombellica
<lb/>[<emph style="it">tr: 
Elliptic, or Bombellian.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1079" xml:space="preserve">
idem specie: <lb/>
nam hic, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>&gt;</mo><mi>b</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
the same case, for here <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>&gt;</mo><mi>b</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1080" xml:space="preserve">
conversio
<lb/>[<emph style="it">tr: 
conversion
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1081" xml:space="preserve">
idem specie:
<lb/>[<emph style="it">tr: 
the same case.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1082" xml:space="preserve">
recte
<lb/>[<emph style="it">tr: 
recte
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f297" o="297" n="593"/>
<div xml:id="echoid-div176" type="page_commentary" level="2" n="176">
<p>
<s xml:id="echoid-s1083" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1083" xml:space="preserve">
A derivation of the canonical equation arising from setting the product
<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><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>f</mi><mo maxsize="1">)</mo></mstyle></math> equal to zero, with a numerical example. <lb/>
See also Add MS 6783, f. 175 (d.9).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head183" xml:space="preserve">
ag)
</head>
<pb file="add_6783_f297v" o="297v" n="594"/>
<pb file="add_6783_f298" o="298" n="595"/>
<pb file="add_6783_f298v" o="298v" n="596"/>
<pb file="add_6783_f299" o="299" n="597"/>
<pb file="add_6783_f299v" o="299v" n="598"/>
<pb file="add_6783_f300" o="300" n="599"/>
<pb file="add_6783_f300v" o="300v" n="600"/>
<pb file="add_6783_f301" o="301" n="601"/>
<div xml:id="echoid-div177" type="page_commentary" level="2" n="177">
<p>
<s xml:id="echoid-s1085" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1085" xml:space="preserve">
A collection of canonical equations with two positive roots. See Add MS 6783, f. 170 (d.14).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s1087" xml:space="preserve">
duæ primæ coefficentes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>b</mi></mstyle></math> <lb/>
duæ postremæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>c</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
the first two coefficients are <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>; the next two are <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f301v" o="301v" n="602"/>
<pb file="add_6783_f302" o="302" n="603"/>
<div xml:id="echoid-div178" type="page_commentary" level="2" n="178">
<p>
<s xml:id="echoid-s1088" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1088" xml:space="preserve">
A canonical equation with three positive roots. See Add MS 6783, f. 169 (d. 15).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head184" xml:space="preserve">
bb<emph style="super">2</emph>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1090" xml:space="preserve">
Parabolica
<lb/>[<emph style="it">tr: 
A parabolic equation.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f302v" o="302v" n="604"/>
<pb file="add_6783_f303" o="303" n="605"/>
<head xml:id="echoid-head185" xml:space="preserve">
bb<emph style="super">2</emph>
</head>
<pb file="add_6783_f303v" o="303v" n="606"/>
<pb file="add_6783_f304" o="304" n="607"/>
<head xml:id="echoid-head186" xml:space="preserve">
bb<emph style="super">2</emph>
</head>
<pb file="add_6783_f304v" o="304v" n="608"/>
<pb file="add_6783_f305" o="305" n="609"/>
<pb file="add_6783_f305v" o="305v" n="610"/>
<pb file="add_6783_f306" o="306" n="611"/>
<pb file="add_6783_f306v" o="306v" n="612"/>
<pb file="add_6783_f307" o="307" n="613"/>
<pb file="add_6783_f307v" o="307v" n="614"/>
<pb file="add_6783_f308" o="308" n="615"/>
<div xml:id="echoid-div179" type="page_commentary" level="2" n="179">
<p>
<s xml:id="echoid-s1091" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1091" 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>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6783_f308v" o="308v" n="616"/>
<pb file="add_6783_f309" o="309" n="617"/>
<pb file="add_6783_f309v" o="309v" n="618"/>
<pb file="add_6783_f310" o="310" n="619"/>
<pb file="add_6783_f310v" o="310v" n="620"/>
<pb file="add_6783_f311" o="311" n="621"/>
<pb file="add_6783_f311v" o="311v" n="622"/>
<pb file="add_6783_f312" o="312" n="623"/>
<div xml:id="echoid-div180" type="page_commentary" level="2" n="180">
<p>
<s xml:id="echoid-s1093" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1093" xml:space="preserve">
This sheet contains some of Harriot's notes on Stevin, <emph style="it">L'arithmetique pratique</emph> (1585).
Pages 44 to 47 contain the 'Quatriesme Distinction' of Part II, on the Rule Alligation.
The working on this page relates to Examples II and III on page 46:
</s>
<lb/>
<quote xml:lang="fra">
Exemple II. <lb/>
Quelcun à deux sortes de blez, la premiere de 2 ß le boisseau, la seconde de 7 ß le boisseau,
&amp; en veut faire 8 boisseaux chacun de 6 ß. Combien prendra il de chacune sorte? <lb/>
Exemple III. <lb/>
Vn maistre de monnoie à cinq sortes d'or, la premier de 2 Karatz, la seconde de 3 Karatz, la troisieme de 4,
la quatriesme de 11, la cinquiesme de 12; Et veut faire vne masse de 20 marcqs à 5 Karatz;
Combien se prendra de l'or de chascune sorte?
<lb/>[<emph style="it">tr: 
Example II. <lb/>
Someone has two sorts of wheat, the first at 2 ß a bushel, the second at 7 ß a bushel,
and wants 8 bushels at 6 ß. How much must he take of each sort? <lb/>
Example III. <lb/>
A money handler has five sorts of gold, the first of 2 carats, the second of 3 carats, the third of 4,
the fourth of 11, the fifth of 12; and wants to make a weight of 20 marks at 5 carats;
How much gold must he take of each sort?
</emph>]<lb/>
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s1095" xml:space="preserve">
Stevin. <lb/>
pag. 46. <lb/>
pratique
</s>
</p>
<pb file="add_6783_f312v" o="312v" n="624"/>
<pb file="add_6783_f313" o="313" n="625"/>
<pb file="add_6783_f313v" o="313v" n="626"/>
<pb file="add_6783_f314" o="314" n="627"/>
<div xml:id="echoid-div181" type="page_commentary" level="2" n="181">
<p>
<s xml:id="echoid-s1096" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1096" xml:space="preserve">
This sheet contains some of Harriot's notes on Stevin, <emph style="it">L'arithmetique pratique</emph> (1585).
Pages 47 to 121 contain the 'Cinquiesme Distinction' of Part II, on the Rule of Interest.
Page 64 contains a table headed 'Table d'interest simple dommageable, de 12 pour 100'.
Harriot confirms the first figure in the table, 8928571, by dividing 1000000000 by 112.
The page also contains other calculations on interest.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head187" xml:space="preserve">
Interest
</head>
<pb file="add_6783_f314v" o="314v" n="628"/>
<pb file="add_6783_f315" o="315" n="629"/>
<pb file="add_6783_f315v" o="315v" n="630"/>
<pb file="add_6783_f316" o="316" n="631"/>
<pb file="add_6783_f316v" o="316v" n="632"/>
<pb file="add_6783_f317" o="317" n="633"/>
<div xml:id="echoid-div182" type="page_commentary" level="2" n="182">
<p>
<s xml:id="echoid-s1098" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1098" xml:space="preserve">
This sheet contains some of Harriot's notes on Stevin, <emph style="it">L'arithmetique pratique</emph> (1585).
Pages 121 to 131 contain the 'Sixiesme Distinction' of Part II, on the Rule of False.
The working on this page relates to Example V on page 127.
</s>
<lb/>
<quote xml:lang="fra">
Exemple V <lb/>
Ily a vn quadrangle rectangle ABCD, duquel la superfice faict 14, &amp; le costé AB est double au costé AD,
la demande est de combien soit chascun desdicts costez.
<lb/>[<emph style="it">tr: 
Example V <lb/>
There is a rectangle ABCD, of which the surface is 14, and the side AB is double the side AD;
the question is how long is each of the said sides.
</emph>]<lb/>
</quote>
<lb/>
<s xml:id="echoid-s1099" xml:space="preserve">
Harriot works a similar problem for a rectangle with area 96. <lb/>
For another example of Harriot's use of the Rule of False, see Add MS 6783, f. 316.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head188" xml:space="preserve" xml:lang="lat">
Regula falsi
<lb/>[<emph style="it">tr: 
Rule of False
</emph>]<lb/>
</head>
<pb file="add_6783_f317v" o="317v" n="634"/>
<pb file="add_6783_f318" o="318" n="635"/>
<pb file="add_6783_f318v" o="318v" n="636"/>
<div xml:id="echoid-div183" type="page_commentary" level="2" n="183">
<p>
<s xml:id="echoid-s1101" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1101" xml:space="preserve">
See Add MS 6785, f. 350.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s1103" xml:space="preserve">
37. p. 16. Elements
</s>
</p>
<pb file="add_6783_f319" o="319" n="637"/>
<pb file="add_6783_f319v" o="319v" n="638"/>
<pb file="add_6783_f320" o="320" n="639"/>
<pb file="add_6783_f320v" o="320v" n="640"/>
<pb file="add_6783_f321" o="321" n="641"/>
<pb file="add_6783_f321v" o="321v" n="642"/>
<div xml:id="echoid-div184" type="page_commentary" level="2" n="184">
<p>
<s xml:id="echoid-s1104" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1104" xml:space="preserve">
A list of writers on astronomy and trigonometry.
For more detailed references to some of these authors see SOURCES.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s1106" xml:space="preserve">
Menelaus. <lb/>
Ptolomaeus. <lb/>
Theon. <lb/>
Arzahel. <lb/>
Thebit. <lb/>
Geber. <lb/>
Albategnius. <lb/>
Purbachius. <lb/>
Regiomontanus. <lb/>
Copernicus. <lb/>
Maurolicus. <lb/>
Rheticus. <lb/>
Vieta. <lb/>
Bressius. <lb/>
Finkius. <lb/>
Clavius. <lb/>
Lansberg. <lb/>
Ursus. <lb/>
Adrianus Romanus. <lb/>
Rudolph van Collen. <lb/>
Pitiscus
</s>
</p>
<pb file="add_6783_f322" o="322" n="643"/>
<pb file="add_6783_f322v" o="322v" n="644"/>
<pb file="add_6783_f323" o="323" n="645"/>
<pb file="add_6783_f323v" o="323v" n="646"/>
<pb file="add_6783_f324" o="324" n="647"/>
<pb file="add_6783_f324v" o="324v" n="648"/>
<pb file="add_6783_f325" o="325" n="649"/>
<pb file="add_6783_f325v" o="325v" n="650"/>
<pb file="add_6783_f326" o="326" n="651"/>
<pb file="add_6783_f326v" o="326v" n="652"/>
<pb file="add_6783_f327" o="327" n="653"/>
<pb file="add_6783_f327v" o="327v" n="654"/>
<pb file="add_6783_f328" o="328" n="655"/>
<pb file="add_6783_f328v" o="328v" n="656"/>
<pb file="add_6783_f329" o="329" n="657"/>
<pb file="add_6783_f329v" o="329v" n="658"/>
<pb file="add_6783_f330" o="330" n="659"/>
<pb file="add_6783_f330v" o="330v" n="660"/>
<pb file="add_6783_f331" o="331" n="661"/>
<pb file="add_6783_f331v" o="331v" n="662"/>
<pb file="add_6783_f332" o="332" n="663"/>
<pb file="add_6783_f332v" o="332v" n="664"/>
<pb file="add_6783_f333" o="333" n="665"/>
<div xml:id="echoid-div185" type="page_commentary" level="2" n="185">
<p>
<s xml:id="echoid-s1107" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1107" xml:space="preserve">
A <emph style="it">second bimedial</emph> is the square root of a third binome; see Eculid X.56.
</s>
<lb/>
<quote>
X.56 If an area be contained by a rational straight line and the third binomial,
the side of the area is the irrational straight line which is called a second bimedial.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head189" xml:space="preserve" xml:lang="lat">
Cubus ex 2<emph style="super">a</emph> e bin: medijs.
<lb/>[<emph style="it">tr: 
The cube of second bimedial.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1109" xml:space="preserve">
2. ex bin. med
<lb/>[<emph style="it">tr: 
From a second bimedial.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1110" xml:space="preserve">
bin. 3
<lb/>[<emph style="it">tr: 
A third binome.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1111" xml:space="preserve">
Ergo cubus. 2<emph style="super">a</emph>. ex bin: medijs
<lb/>[<emph style="it">tr: 
Therefore a cube from second bimedial.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f333v" o="333v" n="666"/>
<div xml:id="echoid-div186" type="page_commentary" level="2" n="186">
<p>
<s xml:id="echoid-s1112" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1112" xml:space="preserve">
For the definition of binomes of the second kind, see Add MS 6782, f. 267. <lb/>
On this page, Harriot shows that the cube of a binome of the second kind is again a binome of the second kind.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head190" xml:space="preserve" xml:lang="lat">
De cubo binomij 2<emph style="super">i</emph>
<lb/>[<emph style="it">tr: 
On the cube of a second binome
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1114" xml:space="preserve">
bin. 1.
<lb/>[<emph style="it">tr: 
a binome of the first kind.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1115" xml:space="preserve">
cubus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>1</mn><mn>8</mn><mn>2</mn><mn>5</mn><mn>2</mn></mrow></msqrt><mo>+</mo><mn>1</mn><mn>3</mn><mn>5</mn></mstyle></math> bin 2<emph style="super">m</emph>.
<lb/>[<emph style="it">tr: 
the cube <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>1</mn><mn>8</mn><mn>2</mn><mn>5</mn><mn>2</mn></mrow></msqrt><mo>+</mo><mn>1</mn><mn>3</mn><mn>5</mn></mstyle></math> is a binome of the second kind.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1116" xml:space="preserve">
[differentia] 27
<lb/>[<emph style="it">tr: 
a decrease of 27
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f334" o="334" n="667"/>
<div xml:id="echoid-div187" type="page_commentary" level="2" n="187">
<p>
<s xml:id="echoid-s1117" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1117" xml:space="preserve">
A <emph style="it">first bimedial</emph> is the square root of a second binome; see Eculid X.55.
</s>
<lb/>
<quote>
X.55 If an area be contained by a rational straight line and the second binomial,
the side of the area is the irrational straight line which is called a first bimedial.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head191" xml:space="preserve" xml:lang="lat">
Cubus. ex 1<emph style="super">a</emph> bin. med:
<lb/>[<emph style="it">tr: 
The cube of a first bimedial.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1119" xml:space="preserve">
1. ex bin. med:
<lb/>[<emph style="it">tr: 
From a first bimedial.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1120" xml:space="preserve">
bin. 2
<lb/>[<emph style="it">tr: 
A second binome.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1121" xml:space="preserve">
Ergo cubus. 1<emph style="super">a</emph>. ex bin: medij.
<lb/>[<emph style="it">tr: 
Therefore the cube from a first bimedial.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f334v" o="334v" n="668"/>
<div xml:id="echoid-div188" type="page_commentary" level="2" n="188">
<p>
<s xml:id="echoid-s1122" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1122" xml:space="preserve">
For the definition of binomes of the first kind, see Add MS 6782, f. 267. <lb/>
On this page, Harriot shows that the cube of a binome of the first kind is again a binome of the first kind.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head192" xml:space="preserve" xml:lang="lat">
De cubo binomij 1<emph style="super">i</emph>
<lb/>[<emph style="it">tr: 
On the cube of a first binome
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1124" xml:space="preserve">
bin. 1.
<lb/>[<emph style="it">tr: 
a binome of the first kind.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1125" xml:space="preserve">
bin. 1. cubus
<lb/>[<emph style="it">tr: 
the cube of a binome of the first kind.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1126" xml:space="preserve">
ergo binomium 1<emph style="super">m</emph>.
<lb/>[<emph style="it">tr: 
therefore a binome of the first kind.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f335" o="335" n="669"/>
<pb file="add_6783_f335v" o="335v" n="670"/>
<pb file="add_6783_f336" o="336" n="671"/>
<head xml:id="echoid-head193" xml:space="preserve" xml:lang="lat">
Ex bin: med. 5. quadrata sunt bin. 1.
<lb/>[<emph style="it">tr: 
From fifth bimedials come first binomes.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1127" xml:space="preserve">
bin. 1.
<lb/>[<emph style="it">tr: 
a first binome.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1128" xml:space="preserve">
bin. 5 ex bin: med. 1.
<lb/>[<emph style="it">tr: 
A fifth binome from a first bimedial.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f336v" o="336v" n="672"/>
<pb file="add_6783_f337" o="337" n="673"/>
<p xml:lang="lat">
<s xml:id="echoid-s1129" xml:space="preserve">
bin. 3.
<lb/>[<emph style="it">tr: 
A third binome
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f337v" o="337v" n="674"/>
<p xml:lang="lat">
<s xml:id="echoid-s1130" xml:space="preserve">
Bin. 2.
<lb/>[<emph style="it">tr: 
A second binome
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1131" xml:space="preserve">
Radix
<lb/>[<emph style="it">tr: 
Root
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f338" o="338" n="675"/>
<pb file="add_6783_f338v" o="338v" n="676"/>
<pb file="add_6783_f339" o="339" n="677"/>
<p xml:lang="lat">
<s xml:id="echoid-s1132" xml:space="preserve">
Bin. 2.
<lb/>[<emph style="it">tr: 
A second binome
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f339v" o="339v" n="678"/>
<p xml:lang="lat">
<s xml:id="echoid-s1133" xml:space="preserve">
bin: 1.
<lb/>[<emph style="it">tr: 
A first binome
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1134" xml:space="preserve">
radix
<lb/>[<emph style="it">tr: 
root
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f340" o="340" n="679"/>
<pb file="add_6783_f340v" o="340v" n="680"/>
<pb file="add_6783_f341" o="341" n="681"/>
<pb file="add_6783_f341v" o="341v" n="682"/>
<div xml:id="echoid-div189" type="page_commentary" level="2" n="189">
<p>
<s xml:id="echoid-s1135" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1135" xml:space="preserve">
Here Harriot creates a systematic listing of the first 53 binomes of the first kind,
using his own formula <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>d</mi></mrow></msqrt></mstyle></math>.
For his derivation of this formula, see Add MS 6783, f. 356v. <lb/>
He also shows how second binomes can be obtained from first binomes by multiplication.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s1137" xml:space="preserve">
Binomia prima
<lb/>[<emph style="it">tr: 
A first binome
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1138" xml:space="preserve">
Binomia 2<emph style="super">a</emph>
<lb/>[<emph style="it">tr: 
A second binome
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1139" xml:space="preserve">
fiunt binomia 2<emph style="super">a</emph> <emph style="super">et 3<emph style="super">a</emph></emph> ex primis.
<lb/>[<emph style="it">tr: 
Second and third binomes arise from first binomes.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f342" o="342" n="683"/>
<pb file="add_6783_f342v" o="342v" n="684"/>
<div xml:id="echoid-div190" type="page_commentary" level="2" n="190">
<p>
<s xml:id="echoid-s1140" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1140" xml:space="preserve">
Here Harriot creates a systematic listing of the first few binomes of the fourth kind,
using his own formula <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi><mo>-</mo><mi>c</mi><mi>d</mi></mrow></msqrt></mstyle></math>.
For his derivation of this formula, see Add MS 6783, f. 355v. <lb/>
He also shows how fifth and sixth binomes can be obtained from fourth binomes by multiplication.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s1142" xml:space="preserve">
Binomia quarta
<lb/>[<emph style="it">tr: 
A fourth binome
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1143" xml:space="preserve">
Binomia 5<emph style="super">a</emph> <emph style="super">et 6<emph style="super">a</emph></emph> fiunt ex quartis.
<lb/>[<emph style="it">tr: 
Fifth and sixth binomes arise from fourth binomes
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1144" xml:space="preserve">
binomium 6.
<lb/>[<emph style="it">tr: 
a sixth binome
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1145" xml:space="preserve">
bin. 5.
<lb/>[<emph style="it">tr: 
a fifth binome
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1146" xml:space="preserve">
ex quintis fiunt 6<emph style="super">a</emph>
<lb/>[<emph style="it">tr: 
from fifth binomes come sixth binomes
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f343" o="343" n="685"/>
<pb file="add_6783_f343v" o="343v" n="686"/>
<div xml:id="echoid-div191" type="page_commentary" level="2" n="191">
<p>
<s xml:id="echoid-s1147" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1147" xml:space="preserve">
Numerical examples for the formulae derived in the preceding pages.
See also Add MS 6783, f. 344.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head194" xml:space="preserve">
14.)
</head>
<pb file="add_6783_f344" o="344" n="687"/>
<div xml:id="echoid-div192" type="page_commentary" level="2" n="192">
<p>
<s xml:id="echoid-s1149" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1149" xml:space="preserve">
Numerical examples for the formulae derived in the preceding pages.
See also Add MS 6783, f. 343v.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s1151" xml:space="preserve">
verte
<lb/>[<emph style="it">tr: 
turn over
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f344v" o="344v" n="688"/>
<head xml:id="echoid-head195" xml:space="preserve">
13.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1152" xml:space="preserve">
Additio radicum <emph style="st">binomialium</emph> <lb/>
binomij et suum residuum: <lb/>
fit per multiplicatione
<lb/>[<emph style="it">tr: 
Addition of roots of binomes and their residuals, done by multiplication.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f345" o="345" n="689"/>
<pb file="add_6783_f345v" o="345v" n="690"/>
<div xml:id="echoid-div193" type="page_commentary" level="2" n="193">
<p>
<s xml:id="echoid-s1153" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1153" xml:space="preserve">
On this page Harriot again finds the square root of a sixth binome (see Add MS 6783, f. 346v),
this time using the simpler form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>c</mi><mi>d</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>f</mi><mi>g</mi></mrow></msqrt></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head196" xml:space="preserve">
12.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1155" xml:space="preserve">
bin. 6.)
<lb/>[<emph style="it">tr: 
a sixth binome
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f346" o="346" n="691"/>
<pb file="add_6783_f346v" o="346v" n="692"/>
<div xml:id="echoid-div194" type="page_commentary" level="2" n="194">
<p>
<s xml:id="echoid-s1156" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1156" xml:space="preserve">
On this page, Harriot finds the root of a sixth binome (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>f</mi><mi>g</mi><mo>+</mo><mi>b</mi><mi>d</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>f</mi><mi>g</mi></mrow></msqrt></mstyle></math>)
by the method shown on Add MS 6783, f. 9, and uses his result to find the square roots of
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>2</mn><mn>4</mn></mrow></msqrt><mo>+</mo><msqrt><mrow><mn>8</mn></mrow></msqrt></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>2</mn><mn>0</mn></mrow></msqrt><mo>+</mo><msqrt><mrow><mn>1</mn><mn>2</mn></mrow></msqrt></mstyle></math>.
The first of these is what Harriot sometimes refers to as bin. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mo>ʹ</mo></mstyle></math>, since <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>4</mn><mo>-</mo><mn>8</mn><mo>=</mo><mn>1</mn><mn>6</mn></mstyle></math>, a square.
The second is what he sometimes refers to as bin. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mo>ʺ</mo></mstyle></math>, since <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>0</mn><mo>-</mo><mn>1</mn><mn>2</mn><mo>=</mo><mn>8</mn></mstyle></math>, a non-square.
In the first case, he notes that the root is a sum of a binome and an apotome of the fifth kind;
in the second case, the root is a sum of a binome and an apotome of the sixth kind. <lb/>
For further results of this kind see Add MS 6783, f. 362v.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head197" xml:space="preserve">
11.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1158" xml:space="preserve">
bin. 6.)
<lb/>[<emph style="it">tr: 
a sixth binome
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1159" xml:space="preserve">
bin.5. ap. 5. bin. 6.
<lb/>[<emph style="it">tr: 
a fifth binome, a fifth apotome, a sixth binome
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1160" xml:space="preserve">
bin.6. Ap. 6.
<lb/>[<emph style="it">tr: 
a sixth binome, a sixth apotome
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f347" o="347" n="693"/>
<pb file="add_6783_f347v" o="347v" n="694"/>
<div xml:id="echoid-div195" type="page_commentary" level="2" n="195">
<p>
<s xml:id="echoid-s1161" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1161" xml:space="preserve">
On this page, Harriot finds the root of a fifth binome (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mo>+</mo><mi>c</mi><mi>d</mi></mrow></msqrt><mo>+</mo><mi>b</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mo>+</mo><mi>d</mi><mi>d</mi></mrow></msqrt><mo>+</mo><mi>b</mi></mstyle></math>)
by the method shown on Add MS 6783, f. 9, and uses his result to find the square roots of
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>8</mn></mrow></msqrt><mo>+</mo><mn>2</mn></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>1</mn><mn>2</mn></mrow></msqrt><mo>+</mo><mn>2</mn></mstyle></math>.
The first of these is what Harriot sometimes refers to as bin. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mo>ʹ</mo></mstyle></math>, since <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>8</mn><mo>-</mo><mrow><msup><mn>2</mn><mn>2</mn></msup></mrow><mo>=</mo><mn>4</mn></mstyle></math>, a square.
The second is what he sometimes refers to as bin. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mo>ʺ</mo></mstyle></math>, since <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>2</mn><mo>-</mo><mrow><msup><mn>2</mn><mn>2</mn></msup></mrow><mo>=</mo><mn>8</mn></mstyle></math>, a non-square
(it is impossible to see whether he does so here because the edge of the page is lost in the binding).
In the first case, he notes that the root is a sum of a binome and an apotome of the fifth kind;
in the second case, the root is a sum of a binome and an apotome of the sixth kind. <lb/>
For further results of this kind see Add MS 6783, f. 362v.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head198" xml:space="preserve">
10.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1163" xml:space="preserve">
Bin. 5.)
<lb/>[<emph style="it">tr: 
a fifth binome
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1164" xml:space="preserve">
bin.5. ap. 5. bin. 5.
<lb/>[<emph style="it">tr: 
a fifth binome, a fifth apotome, a fifth binome
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1165" xml:space="preserve">
bin.6. ap. 6. bin. 6.
<lb/>[<emph style="it">tr: 
a sixth binome, a sixth apotome, a sixth binome
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f348" o="348" n="695"/>
<pb file="add_6783_f348v" o="348v" n="696"/>
<div xml:id="echoid-div196" type="page_commentary" level="2" n="196">
<p>
<s xml:id="echoid-s1166" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1166" xml:space="preserve">
On this page, Harriot finds the root of a fourth binome (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi><mo>-</mo><mi>c</mi><mi>d</mi></mrow></msqrt></mstyle></math>)
by the method shown on Add MS 6783, f. 9, and uses his result to find the square roots of
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mo>+</mo><msqrt><mrow><mn>8</mn></mrow></msqrt></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mo>+</mo><msqrt><mrow><mn>2</mn><mn>8</mn></mrow></msqrt></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head199" xml:space="preserve">
9.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1168" xml:space="preserve">
Bin. 4.)
<lb/>[<emph style="it">tr: 
a fourth binome
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f349" o="349" n="697"/>
<pb file="add_6783_f349v" o="349v" n="698"/>
<div xml:id="echoid-div197" type="page_commentary" level="2" n="197">
<p>
<s xml:id="echoid-s1169" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1169" xml:space="preserve">
On this page, Harriot writes down the root of a third binome
(<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mfrac><mrow><mi>b</mi><mi>b</mi><mi>f</mi><mi>f</mi></mrow><mrow><mi>c</mi><mi>g</mi></mrow></mfrac><mo>+</mo><msqrt><mrow><mfrac><mrow><mi>b</mi><mi>b</mi><mi>f</mi><mi>f</mi></mrow><mrow><mi>c</mi><mi>g</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>d</mi><mi>f</mi><mi>f</mi></mrow><mrow><mi>c</mi><mi>g</mi></mrow></mfrac></mrow></msqrt></mrow></msqrt></mstyle></math>).
using the formula derived on Add MS 6788, f. 15 (and elsewhere),
and uses his result to find the square roots of
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>3</mn><mn>2</mn></mrow></msqrt><mo>+</mo><msqrt><mrow><mn>2</mn><mn>4</mn></mrow></msqrt></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>1</mn><mn>0</mn><mn>8</mn></mrow></msqrt><mo>+</mo><msqrt><mrow><mn>9</mn><mn>6</mn></mrow></msqrt></mstyle></math>.
In each case the square root is a second bimedial (see Add MS 6783, f. 356v). <lb/>
In the lower half of the page, he demonstrates algebraically that the square root is a second bimedial,
according to Euclid's definition in X.38 (see Add MS 6783, f. 356v),
because the product of the two parts is a medial (or square root).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head200" xml:space="preserve">
8.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1171" xml:space="preserve">
bin. 3.)
<lb/>[<emph style="it">tr: 
a third binome
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1172" xml:space="preserve">
radix
<lb/>[<emph style="it">tr: 
root
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1173" xml:space="preserve">
medium
<lb/>[<emph style="it">tr: 
mean
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1174" xml:space="preserve">
ex binis medijs<lb/>
secunda.
<lb/>[<emph style="it">tr: 
second of the two means.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1175" xml:space="preserve">
rectangularum partium
<lb/>[<emph style="it">tr: 
parts of the rectangle
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1176" xml:space="preserve">
medium
<lb/>[<emph style="it">tr: 
mean
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1177" xml:space="preserve">
medium
<lb/>[<emph style="it">tr: 
mean
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f350" o="350" n="699"/>
<pb file="add_6783_f350v" o="350v" n="700"/>
<div xml:id="echoid-div198" type="page_commentary" level="2" n="198">
<p>
<s xml:id="echoid-s1178" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1178" xml:space="preserve">
Here Harriot demonstrates algebraically that the expression at the top of the page is a first bimedial,
according to Euclid's definition in X.37 (see Add MS 6783, f. 356v),
because the product of the two parts is rational.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head201" xml:space="preserve">
7.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1180" xml:space="preserve">
bimedialis 1<emph style="super">a</emph>
<lb/>[<emph style="it">tr: 
a first bimedial
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1181" xml:space="preserve">
rectangulum <lb/>
ex partis
<lb/>[<emph style="it">tr: 
rectangle from the parts
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1182" xml:space="preserve">
aliter
<lb/>[<emph style="it">tr: 
another way
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1183" xml:space="preserve">
vide sequentem chartam
<lb/>[<emph style="it">tr: 
see the next sheet
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1184" xml:space="preserve">
rectangulum <lb/>
ex mediis
<lb/>[<emph style="it">tr: 
rectangle from the means
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f351" o="351" n="701"/>
<pb file="add_6783_f351v" o="351v" n="702"/>
<div xml:id="echoid-div199" type="page_commentary" level="2" n="199">
<p>
<s xml:id="echoid-s1185" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1185" xml:space="preserve">
The note 'prop 56.1' at the top of this page is a reference to Euclid's <emph style="it">Elements</emph>,
Book X, Proposition 56. In modern editions the relevant proposition is X.55,
but Harriot's numbering matches that of both Commandino and Clavius.
</s>
<lb/>
<quote>
If an area is contained by a rational straight line and the second binomial,
then the side of the area is the irrational straight line called a first bimedial. <lb/>
</quote>
<lb/>
<s xml:id="echoid-s1186" xml:space="preserve">
In modern terms, we may interpret this as saying that the square root of a second binome is a first bimedial.
On this page, Harriot finds the root of a second binome (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mfrac><mrow><mi>b</mi><mi>b</mi><mi>f</mi><mi>f</mi></mrow><mrow><mi>b</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>d</mi></mrow></mfrac></mrow></msqrt><mo>+</mo><mi>f</mi></mstyle></math>)
by the method shown on Add MS 6783, f. 9, and uses his result to find the square roots of
<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>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>4</mn><mo>+</mo><msqrt><mrow><mn>1</mn><mn>8</mn><mn>0</mn></mrow></msqrt></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mo>+</mo><msqrt><mrow><mn>8</mn></mrow></msqrt></mstyle></math>.
In each case the square root is a first bimedial (see Add MS 6783, f. 356v).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head202" xml:space="preserve" xml:lang="lat">
6.) prop. 56.1.
<lb/>[<emph style="it">tr: 
Proposition 56
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1188" xml:space="preserve">
bin. 2.)
<lb/>[<emph style="it">tr: 
a second binome
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1189" xml:space="preserve">
Radix
<lb/>[<emph style="it">tr: 
root
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1190" xml:space="preserve">
Quæ ex binis <lb/>
mediis prima.
<lb/>[<emph style="it">tr: 
Which is the first of the two means.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f352" o="352" n="703"/>
<pb file="add_6783_f352v" o="352v" n="704"/>
<div xml:id="echoid-div200" type="page_commentary" level="2" n="200">
<p>
<s xml:id="echoid-s1191" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1191" xml:space="preserve">
In the upper half of this page, Harriot finds the square of a second binome,
and demonstrates that the square is a first binome. <lb/>
In the lower right hand corner of the page, he squares a first binome,
again demonstrating that the square is a first binome. <lb/>
See also Add MS Add MS 6782, f. 267.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head203" xml:space="preserve" xml:lang="lat">
5.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1193" xml:space="preserve">
bin. 2:
<lb/>[<emph style="it">tr: 
a second binome
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1194" xml:space="preserve">
minimum <lb/>
superficies
<lb/>[<emph style="it">tr: 
minimum surface
</emph>]<lb/>
<lb/>[<emph style="it">tr: 
a second binome
</emph>]<lb/>
[<emph style="it">Note: 
Harriot has already noted on Add MS 6783, f. 356v that <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>
is the smallest binome that can represent an area.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1195" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi><mi>d</mi><mi>d</mi></mstyle></math> differentia quadratorum
<lb/>[<emph style="it">tr: 
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi><mi>d</mi><mi>d</mi></mstyle></math> difference of the squares
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1196" xml:space="preserve">
bin. 1. lineare
<lb/>[<emph style="it">tr: 
a first linear binome
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1197" xml:space="preserve">
bin. 1. superficiale
<lb/>[<emph style="it">tr: 
a first plane binome
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f353" o="353" n="705"/>
<pb file="add_6783_f353v" o="353v" n="706"/>
<div xml:id="echoid-div201" type="page_commentary" level="2" n="201">
<p>
<s xml:id="echoid-s1198" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1198" xml:space="preserve">
The note 'pro: 55' at the top of this page is a reference to Eculid's <emph style="it">Elements</emph>,
Book X, Proposition 55. In modern editions the relevant proposition is X.54,
but Harriot's numbering matches that of both Commandino and Clavius.
</s>
<lb/>
<quote>
If an area be contained by a rational straight line and the first binomial,
the side of the area is the irrational straight line which is called binomial. <lb/>
</quote>
<lb/>
<s xml:id="echoid-s1199" xml:space="preserve">
In modern terms, we may interpret this as saying that the square root of a first binome is again a binome.
On this page, Harriot finds the root of a first binome (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>d</mi></mrow></msqrt></mstyle></math>)
by the method shown on Add MS 6783, f. 9, and shows how the result can be used to find the square roots of
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>1</mn><mn>2</mn></mrow></msqrt></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>7</mn><mn>2</mn></mrow></msqrt><mo>+</mo><mn>8</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>1</mn><mn>8</mn></mrow></msqrt><mo>+</mo><mn>4</mn></mstyle></math>, and so on.
In each case the square root is a first bimedial.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head204" xml:space="preserve" xml:lang="lat">
4.) pro: 55.
<lb/>[<emph style="it">tr: 
proposition 55
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1201" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>d</mi></mrow></msqrt></mstyle></math> bin 1<emph style="super">a</emph>
<lb/>[<emph style="it">tr: 
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>d</mi></mrow></msqrt></mstyle></math> a first binome
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1202" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>d</mi><mi>d</mi><mi>c</mi><mi>c</mi></mrow></msqrt></mstyle></math> bin 1<emph style="super">m</emph>
<lb/>[<emph style="it">tr: 
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>d</mi><mi>d</mi><mi>c</mi><mi>c</mi></mrow></msqrt></mstyle></math> a first binome
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1203" xml:space="preserve">
radix
<lb/>[<emph style="it">tr: 
root
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f354" o="354" n="707"/>
<pb file="add_6783_f354v" o="354v" n="708"/>
<div xml:id="echoid-div202" type="page_commentary" level="2" n="202">
<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 derives general formulae for binomes of the fifth sixth kind. <lb/>
The upper half of the page, below the number '5.)', contains several versions of a formula for fifth binomes,
the simplest of which, marked 'melius' (best), is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mo>+</mo><mi>c</mi><mi>d</mi></mrow></msqrt><mo>+</mo><mi>b</mi></mstyle></math>.
Note, however, the special case <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mo>+</mo><mi>c</mi><mi>c</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi></mrow></msqrt></mstyle></math>,
where the difference between the square of the two terms is itself a square.
Elsewhere, Harriot denotes this special kind (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mo>+</mo><mi>c</mi><mi>c</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi></mrow></msqrt></mstyle></math>) as bin. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mo>ʹ</mo></mstyle></math>
and the more general kind (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mo>+</mo><mi>c</mi><mi>d</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi></mrow></msqrt></mstyle></math>) as bin. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mo>ʺ</mo></mstyle></math>. <lb/>
The lower half of the page, below the number '6.)', contains several versions of a formula for sixth binomes,
the simplest and most general of which is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>f</mi><mi>g</mi><mo>+</mo><mi>c</mi><mi>d</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>f</mi><mi>g</mi></mrow></msqrt></mstyle></math>.
Note, however, as for fifthe binomes, the special case <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>f</mi><mi>g</mi><mo>+</mo><mi>c</mi><mi>c</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>f</mi><mi>g</mi></mrow></msqrt></mstyle></math>,
where the difference between the square of the two terms is itself a square.
Elsewhere, Harriot denotes this special kind (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>f</mi><mi>g</mi><mo>+</mo><mi>c</mi><mi>c</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>f</mi><mi>g</mi></mrow></msqrt></mstyle></math>) as bin. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mo>ʹ</mo></mstyle></math>
and the more general kind (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>f</mi><mi>g</mi><mo>+</mo><mi>c</mi><mi>d</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>f</mi><mi>g</mi></mrow></msqrt></mstyle></math>) as bin. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mo>ʺ</mo></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head205" xml:space="preserve" xml:lang="lat">
3.) De binomijs
<lb/>[<emph style="it">tr: 
On binomes
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1206" xml:space="preserve">
melius
<lb/>[<emph style="it">tr: 
better
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1207" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>+</mo><mi>c</mi><mi>c</mi></mstyle></math> sit non quadratus
<lb/>[<emph style="it">tr: 
let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>+</mo><mi>c</mi><mi>c</mi></mstyle></math> not be a square
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1208" xml:space="preserve">
5.)
[<emph style="it">Note: 
General formulae for binomes of the fifth kind.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1209" xml:space="preserve">
melius
<lb/>[<emph style="it">tr: 
better
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1210" xml:space="preserve">
AB quadratus
<lb/>[<emph style="it">tr: 
AB is a square
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1211" xml:space="preserve">
AC non quad et pars
<lb/>[<emph style="it">tr: 
AC is not a square, and a part
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1212" xml:space="preserve">
CB non quad <emph style="super">altera</emph>
<lb/>[<emph style="it">tr: 
CB is not a square, another part
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1213" xml:space="preserve">
D non quad et ad <lb/>
AC ut non quad <lb/>
et AB.
<lb/>[<emph style="it">tr: 
D is not a square, and is not as a square to AC or AB.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1214" xml:space="preserve">
E rationalis long.
<lb/>[<emph style="it">tr: 
E is a rational length
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1215" xml:space="preserve">
6.)
[<emph style="it">Note: 
General formulae for binomes of the sixth kind.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1216" xml:space="preserve">
aliter
<lb/>[<emph style="it">tr: 
another way
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1217" xml:space="preserve">
Aliter
<lb/>[<emph style="it">tr: 
Another way
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f355" o="355" n="709"/>
<pb file="add_6783_f355v" o="355v" n="710"/>
<div xml:id="echoid-div203" type="page_commentary" level="2" n="203">
<p>
<s xml:id="echoid-s1218" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1218" xml:space="preserve">
On this page, Harriot derives general formulae for binomes of the third and fourth kind. <lb/>
The upper half of the page, below the number 3.), contains several versions of a formula for third binomes,
the simplest and most general of which is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mi>c</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi><mi>c</mi><mo>-</mo><mi>d</mi><mi>d</mi><mi>c</mi></mrow></msqrt></mstyle></math>. <lb/>
The lower half of the page, below the number 4.), contains several versions of a formula for fourth binomes,
the simplest of which, marked 'melius' (best), is in the bottom right hand corner of the page,
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi><mo>-</mo><mi>c</mi><mi>d</mi></mrow></msqrt></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head206" xml:space="preserve" xml:lang="lat">
2.) De binomijs
<lb/>[<emph style="it">tr: 
On binomes
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1220" xml:space="preserve">
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math> quadratus
<lb/>[<emph style="it">tr: 
let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math> be a square
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1221" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>d</mi></mstyle></math> <emph style="super">sit</emph> non quadratus
<lb/>[<emph style="it">tr: 
let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>d</mi></mstyle></math> not be a square
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1222" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>g</mi></mstyle></math> <emph style="super">sit</emph> non quadratus et ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>d</mi></mstyle></math> non ut quadratus ad
<emph style="super">quadratum</emph> <lb/>
ita erit ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>g</mi></mstyle></math> not be a square and to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>d</mi></mstyle></math> not as a square to a square; thus it will be to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math>
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1223" xml:space="preserve">
3.)
[<emph style="it">Note: 
General formulae for binomes of the third kind.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1224" xml:space="preserve">
melius
<lb/>[<emph style="it">tr: 
better
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1225" xml:space="preserve">
<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>c</mi><mo>+</mo><mi>c</mi><mi>c</mi></mstyle></math> quadratus
<lb/>[<emph style="it">tr: 
<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>c</mi><mo>+</mo><mi>c</mi><mi>c</mi></mstyle></math> is a square
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1226" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi></mstyle></math> sit non quadratus
<lb/>[<emph style="it">tr: 
let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi></mstyle></math> not be a square
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1227" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi></mstyle></math> sit non quadratus
<lb/>[<emph style="it">tr: 
let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi></mstyle></math> not be a square
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1228" xml:space="preserve">
4.)
[<emph style="it">Note: 
General formulae for binomes of the fourth kind.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1229" xml:space="preserve">
Ergo, sequentes species <lb/>
sunt æque <lb/>
universalis.
<lb/>[<emph style="it">tr: 
Therefore the following forms are equally general.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1230" xml:space="preserve">
Item AB <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>c</mi><mo>+</mo><mi>c</mi><mi>c</mi></mstyle></math>
<lb/>[<emph style="it">tr: 
Similarly <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>c</mi><mo>+</mo><mi>c</mi><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1231" xml:space="preserve">
AC <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> non quad
<lb/>[<emph style="it">tr: 
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> is not a square
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1232" xml:space="preserve">
CB <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></mstyle></math> non quad
<lb/>[<emph style="it">tr: 
<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></mstyle></math> is not a square
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1233" xml:space="preserve">
melius
<lb/>[<emph style="it">tr: 
better
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f356" o="356" n="711"/>
<pb file="add_6783_f356v" o="356v" n="712"/>
<div xml:id="echoid-div204" type="page_commentary" level="2" n="204">
<p>
<s xml:id="echoid-s1234" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1234" xml:space="preserve">
This is the first of a set of 14 numbered pages on binomes.
The six types of binome are defined in Book X of Euclid's <emph style="it">Elements</emph>
in definitions preceding Proposition 48, in the following way.
</s>
<lb/>
<quote>
1. Given a rational straight line and a binomial, divided into its terms,
such that the square on the greater term is greater than the square on the lesser
by the square on a straight line commensurable in length with the greater,
then, if the greater term is commensurable in length with the rational straight line set out,
let the whole be called a first binomial straight line. <lb/>
2. But if the lesser term is commensurable in length with the rational straight line set out,
let the whole be called a second binomial. <lb/>
3. And if neither of the terms is commensurable in length with the rational straight line set out,
let the whole be called a third binomial. <lb/>
4. Again, if the square on the greater term is greater than the square on the lesser
by the square on a straight line incommensurable in length with the greater,
then, if the greater term is commensurable in length with the rational straight line set out,
let the whole be called a fourth binomial. <lb/>
5. If the lesser, a fifth binomial. <lb/>
6. And if neither, a sixth binomial.
</quote>
<lb/>
<s xml:id="echoid-s1235" xml:space="preserve">
These definitions may be written in modern notation 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/>
</s>
<lb/>
<s xml:id="echoid-s1236" xml:space="preserve">
Harriot also investigated two further kinds of irrational quantities defined by Euclid:
first and second bimedials. These are defined in Euclid X.37 and X.38.
</s>
<lb/>
<quote>
X.37 If two medial straight lines commensurable in quare only and containing a rational rectangle be added together,
the whole is irrational; and let it be called a <emph style="it">first bimedial</emph> straight line. <lb/>
X.38 If two medial straight lines commensurable in quare only and containing a medial rectangle be added together,
the whole is irrational; and let it be called a <emph style="it">second bimedial</emph> straight line.
</quote>
<lb/>
<s xml:id="echoid-s1237" xml:space="preserve">
On this page, Harriot derives general formulae for binomes of the first and second kind. <lb/>
Three lines after the word 'Aliter' on the right is his formula for first binomes:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>d</mi></mrow></msqrt></mstyle></math>. <lb/>
On the lower half of the page, below the number 2.) is his formula for second binomes:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mfrac><mrow><mi>b</mi><mi>b</mi><mi>f</mi><mi>f</mi></mrow><mrow><mi>b</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>d</mi></mrow></mfrac><mo>+</mo><mi>f</mi></mrow></msqrt></mstyle></math>. <lb/>
In each case he gives several numerical exmples. <lb/>
For further work on binomes see other nearby sheets, Add MS 6783, f. 337 to f. 342.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head207" xml:space="preserve" xml:lang="lat">
1.) De numeris planis <lb/>
et binomialium linearum speciebus. <lb/>[<emph style="it">tr: 
On plane numbers and types of linear binomials.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1239" xml:space="preserve">
Factus æquatur proportionalibus <lb/>
est quadratus.
<lb/>[<emph style="it">tr: 
The product of equal proportionals is a square.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1240" xml:space="preserve">
Plani similes
<lb/>[<emph style="it">tr: 
Similar planes
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1241" xml:space="preserve">
Sit, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>d</mi></mstyle></math>, <emph style="super">sit</emph> non quadratus.
<lb/>[<emph style="it">tr: 
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>d</mi></mstyle></math> be a non-square.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1242" xml:space="preserve">
<lb/>[<emph style="it">tr: 
This part of the page contains general formulae for binomes of the first kind.
</emph>]<lb/>
* Aliter.
<lb/>[<emph style="it">tr: 
Another way.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1243" xml:space="preserve">
bin <lb/>
minima <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mo>+</mo><msqrt><mrow><mn>3</mn></mrow></msqrt></mstyle></math>.
<lb/>[<emph style="it">tr: 
least binome: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mo>+</mo><msqrt><mrow><mn>3</mn></mrow></msqrt></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1244" xml:space="preserve">
minimum superfic. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mo>+</mo><msqrt><mrow><mn>8</mn></mrow></msqrt></mstyle></math>.
<lb/>[<emph style="it">tr: 
minimum surface: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mo>+</mo><msqrt><mrow><mn>8</mn></mrow></msqrt></mstyle></math>.
</emph>]<lb/>
[<emph style="it">Note: 
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mo>+</mo><msqrt><mrow><mn>8</mn></mrow></msqrt></mstyle></math> is the square of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>+</mo><mn>1</mn></mstyle></math>, the smallest possible binome;
it therefore represents the smallest surface that can be constructed from two linear binomes.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1245" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>+</mo><mn>1</mn></mstyle></math> 5 bin. minimum absolute
<lb/>[<emph style="it">tr: 
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>+</mo><mn>1</mn></mstyle></math>, a binome of the fifth kind, the absolute least.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1246" xml:space="preserve">
2.)
[<emph style="it">Note: 
General formulae for binomes of the second kind.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1247" xml:space="preserve">
differentia
<lb/>[<emph style="it">tr: 
difference
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1248" xml:space="preserve">
optime
<lb/>[<emph style="it">tr: 
best
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1249" xml:space="preserve">
Aliter
<lb/>[<emph style="it">tr: 
Another way
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1250" xml:space="preserve">
differentia
<lb/>[<emph style="it">tr: 
difference
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f357" o="357" n="713"/>
<pb file="add_6783_f357v" o="357v" n="714"/>
<div xml:id="echoid-div205" type="page_commentary" level="2" n="205">
<p>
<s xml:id="echoid-s1251" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1251" xml:space="preserve">
An attempt to find the cube root of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>6</mn><mo>+</mo><msqrt><mrow><mn>6</mn><mn>7</mn><mn>5</mn></mrow></msqrt></mstyle></math>, a binome of the first kind,
by the rule given in Add MS 6783, f. 395v.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head208" xml:space="preserve" xml:lang="lat">
Extractio radicis cubicæ <lb/>
e solido binomio.
<lb/>[<emph style="it">tr: 
Extraction of the cube root of a solid binome.
</emph>]<lb/>
</head>
<pb file="add_6783_f358" o="358" n="715"/>
<pb file="add_6783_f358v" o="358v" n="716"/>
<div xml:id="echoid-div206" type="page_commentary" level="2" n="206">
<p>
<s xml:id="echoid-s1253" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1253" xml:space="preserve">
This folio shows how to find the individual quantities <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>a</mi></mrow></msqrt></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>c</mi><mi>e</mi></mrow></msqrt></mstyle></math>
given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>a</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>c</mi><mi>e</mi></mrow></msqrt></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi><mo>-</mo><mi>c</mi><mi>e</mi></mstyle></math>. <lb/>
(This is possibly related to Add MS 6783, f. 792,
where Harriot is also working with the sum of two roots (there <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>f</mi><mi>f</mi><mi>f</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>g</mi><mi>g</mi><mi>g</mi></mrow></msqrt></mstyle></math>)
and the difference of their squares (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>f</mi><mi>f</mi><mo>-</mo><mi>g</mi><mi>g</mi><mi>g</mi></mstyle></math>).)
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head209" xml:space="preserve" xml:lang="lat">
Lemma: ad extractionem radicis cubicæ e binomio solido.
<lb/>[<emph style="it">tr: 
Lemma for the extraction of cube roots of solid binomes.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1255" xml:space="preserve">
Dantur:
<lb/>[<emph style="it">tr: 
Given:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1256" xml:space="preserve">
Quæritur
<lb/>[<emph style="it">tr: 
sought
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1257" xml:space="preserve">
Illustratio per numeros.
<lb/>[<emph style="it">tr: 
An illustration in numbers
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1258" xml:space="preserve">
Canones.
<lb/>[<emph style="it">tr: 
Rules.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f359" o="359" n="717"/>
<pb file="add_6783_f359v" o="359v" n="718"/>
<pb file="add_6783_f360" o="360" n="719"/>
<pb file="add_6783_f360v" o="360v" n="720"/>
<div xml:id="echoid-div207" type="page_commentary" level="2" n="207">
<p>
<s xml:id="echoid-s1259" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1259" xml:space="preserve">
Some numerical examples of the rules for square roots of sums,
possibly continuing from the bottom of Add MS 6783, f. 362v,
where both the content and the writing style are similar.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6783_f361" o="361" n="721"/>
<div xml:id="echoid-div208" type="page_commentary" level="2" n="208">
<p>
<s xml:id="echoid-s1261" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1261" xml:space="preserve">
Some further numerical examples of the rules for square roots of first and second binomes,
apparently continued from Add MS 6783, f. 360v
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6783_f361v" o="361v" n="722"/>
<pb file="add_6783_f362" o="362" n="723"/>
<pb file="add_6783_f362v" o="362v" n="724"/>
<div xml:id="echoid-div209" type="page_commentary" level="2" n="209">
<p>
<s xml:id="echoid-s1263" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1263" xml:space="preserve">
Square roots of binomes by the rule given on Add MS 6788, f. 15 (and elsewhere). <lb/>
The first section demonstrates that the square root of a third binome is
the root of a third binome plus the root of a third apotome. <lb/>
The third section gives the square root of a fourth binome. <lb/>
The fourth section demonstrates that the square root of a fifth binome of the second kind is
the root of a sixth binome of the first kind plus the root of a sixth apotome of the first kind. <lb/>
The fifth section demonstrates that the square root of a fifth binome of the first kind is
the root of a fifth binome of the first kind plus the root a fifth apotome of the first kind. <lb/>
The sixth section demonstrates that the square root of a sixth binome of the first kind is
the root of a fifth binome of the second kind plus the root a fifth apotome of the second kind,
and that the square root of a sixth binome of the second kind is
the root of a sixth binome of the second kind plus the root a sixth apotome of the second kind. <lb/>
The seventh section demonstrates again that the square root of a sixth binome of the first kind is
the root of a fifth binome of the second kind plus the root a fifth apotome of the second kind,
and that the square root of a sixth binome of the second kind is
the root of a sixth binome of the second kind plus the root a sixth apotome of the second kind. <lb/>
For Harriot's two kind of fifth and sixth binomes, see Add MS 6784, 216.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6783_f363" o="363" n="725"/>
<pb file="add_6783_f363v" o="363v" n="726"/>
<pb file="add_6783_f364" o="364" n="727"/>
<pb file="add_6783_f364v" o="364v" n="728"/>
<pb file="add_6783_f365" o="365" n="729"/>
<pb file="add_6783_f365v" o="365v" n="730"/>
<pb file="add_6783_f366" o="366" n="731"/>
<pb file="add_6783_f366v" o="366v" n="732"/>
<pb file="add_6783_f367" o="367" n="733"/>
<pb file="add_6783_f367v" o="367v" n="734"/>
<pb file="add_6783_f368" o="368" n="735"/>
<pb file="add_6783_f368v" o="368v" n="736"/>
<pb file="add_6783_f369" o="369" n="737"/>
<pb file="add_6783_f369v" o="369v" n="738"/>
<pb file="add_6783_f370" o="370" n="739"/>
<pb file="add_6783_f370v" o="370v" n="740"/>
<pb file="add_6783_f371" o="371" n="741"/>
<pb file="add_6783_f371v" o="371v" n="742"/>
<pb file="add_6783_f372" o="372" n="743"/>
<pb file="add_6783_f372v" o="372v" n="744"/>
<pb file="add_6783_f373" o="373" n="745"/>
<pb file="add_6783_f373v" o="373v" n="746"/>
<pb file="add_6783_f374" o="374" n="747"/>
<pb file="add_6783_f374v" o="374v" n="748"/>
<pb file="add_6783_f375" o="375" n="749"/>
<pb file="add_6783_f375v" o="375v" n="750"/>
<pb file="add_6783_f376" o="376" n="751"/>
<pb file="add_6783_f376v" o="376v" n="752"/>
<pb file="add_6783_f377" o="377" n="753"/>
<pb file="add_6783_f377v" o="377v" n="754"/>
<pb file="add_6783_f378" o="378" n="755"/>
<pb file="add_6783_f378v" o="378v" n="756"/>
<pb file="add_6783_f379" o="379" n="757"/>
<pb file="add_6783_f379v" o="379v" n="758"/>
<pb file="add_6783_f380" o="380" n="759"/>
<pb file="add_6783_f380v" o="380v" n="760"/>
<pb file="add_6783_f381" o="381" n="761"/>
<pb file="add_6783_f381v" o="381v" n="762"/>
<pb file="add_6783_f382" o="382" n="763"/>
<pb file="add_6783_f382v" o="382v" n="764"/>
<pb file="add_6783_f383" o="383" n="765"/>
<pb file="add_6783_f383v" o="383v" n="766"/>
<pb file="add_6783_f384" o="384" n="767"/>
<pb file="add_6783_f384v" o="384v" n="768"/>
<pb file="add_6783_f385" o="385" n="769"/>
<pb file="add_6783_f385v" o="385v" n="770"/>
<pb file="add_6783_f386" o="386" n="771"/>
<pb file="add_6783_f386v" o="386v" n="772"/>
<div xml:id="echoid-div210" type="page_commentary" level="2" n="210">
<p>
<s xml:id="echoid-s1265" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1265" xml:space="preserve">
For the definition of binomes of the fifth kind, see Add MS 6782, f. 267. <lb/>
On this page, Harriot shows that the cube of a binome of the fifth kind is again a binome of the fifth kind.
Note his distinction between two types of binome of the fifth kind (see Add MS 6782, f. 267).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head210" xml:space="preserve" xml:lang="lat">
De cubo binomij 5<emph style="super">i</emph> <lb/>
et 5<emph style="super">ii</emph>
<lb/>[<emph style="it">tr: 
On the cube of a fifth binome of the first kind <lb/>
and a fifth binome of the second kind
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1267" xml:space="preserve">
bin. 5.
<lb/>[<emph style="it">tr: 
a binome of the fifth kind.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1268" xml:space="preserve">
bin. 1.
<lb/>[<emph style="it">tr: 
a binome of the first kind.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1269" xml:space="preserve">
ergo cubus:
<lb/>[<emph style="it">tr: 
the cube:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1270" xml:space="preserve">
d[iffe]r[enti]a. q[uadrata]. <lb/>
(64. c[ubus]. est 4
<lb/>[<emph style="it">tr: 
the difference of the squares <lb/>
(64 is the cube of 4
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1271" xml:space="preserve">
bin. 5.
<lb/>[<emph style="it">tr: 
a binome of the fifth kind.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1272" xml:space="preserve">
bin. 1.
<lb/>[<emph style="it">tr: 
a binome of the first kind.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1273" xml:space="preserve">
ergo cubus bin. 5.
<lb/>[<emph style="it">tr: 
therefore the cube is a binome of the fifth kind.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1274" xml:space="preserve">
d[iffe]r[enti]a. quad[rata]. <lb/>
(512 cub[us]
<lb/>[<emph style="it">tr: 
the difference of the squares <lb/>
(512 is a cube
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f387" o="387" n="773"/>
<pb file="add_6783_f387v" o="387v" n="774"/>
<div xml:id="echoid-div211" type="page_commentary" level="2" n="211">
<p>
<s xml:id="echoid-s1275" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1275" xml:space="preserve">
For the definition of binomes of the sixth kind, see Add MS 6782, f. 267. <lb/>
On this page, Harriot shows that the cube of a binome of the sixth kind is again a binome of the sixth kind.
Note his distinction between two types of binome of the sixth kind (see Add MS 6782, f. 267).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head211" xml:space="preserve" xml:lang="lat">
De cubo binomij 6<emph style="super">i</emph> <lb/>
et 6<emph style="super">ii</emph>
<lb/>[<emph style="it">tr: 
On the cube of a sixth binome of the first kind <lb/>
and a sixth binome of the second kind
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1277" xml:space="preserve">
bin. 6.
<lb/>[<emph style="it">tr: 
a binome of the sixth kind.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1278" xml:space="preserve">
bin. 1.
<lb/>[<emph style="it">tr: 
a binome of the first kind.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1279" xml:space="preserve">
ergo cubus. bin. 6.
<lb/>[<emph style="it">tr: 
therefore the cube is a bnome of the sixth kind.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1280" xml:space="preserve">
(dif[ferentia]: q[uadrata]: <lb/>
4096
<lb/>[<emph style="it">tr: 
(the difference of the squares <lb/>
is 4096
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1281" xml:space="preserve">
bin. 6.
<lb/>[<emph style="it">tr: 
a binome of the sixth kind.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1282" xml:space="preserve">
bin. 1.
<lb/>[<emph style="it">tr: 
a binome of the first kind.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1283" xml:space="preserve">
ergo cubus bin. 6.
<lb/>[<emph style="it">tr: 
therefore the cube is a binome of the sixth kind.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1284" xml:space="preserve">
dif[fferentia]: qua[drata]: 512 <lb/>
cub[us]
<lb/>[<emph style="it">tr: 
the difference of the squares 512 <lb/>
is a cube
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f388" o="388" n="775"/>
<pb file="add_6783_f388v" o="388v" n="776"/>
<pb file="add_6783_f389" o="389" n="777"/>
<div xml:id="echoid-div212" type="page_commentary" level="2" n="212">
<p>
<s xml:id="echoid-s1285" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1285" xml:space="preserve">
In this page Harriot offers two proofs of the inequality <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mi>b</mi><mi>c</mi><mi>c</mi><mo>&gt;</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s1287" xml:space="preserve">
Aliud lemma
<lb/>[<emph style="it">tr: 
Another lemma
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1288" xml:space="preserve">
Sit.
<lb/>[<emph style="it">tr: 
Let
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1289" 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-s1290" xml:space="preserve">
ponatur:
<lb/>[<emph style="it">tr: 
suppose it is true.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1291" xml:space="preserve">
Ergo:
<lb/>[<emph style="it">tr: 
Therefore:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1292" xml:space="preserve">
Ergo:
<lb/>[<emph style="it">tr: 
Therefore:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1293" xml:space="preserve">
Et ita est: <lb/>
per lemma (versa pagina)
<lb/>[<emph style="it">tr: 
That is (by the lemma on the other side of the page):
</emph>]<lb/>
[<emph style="it">Note: 
The other side of the page is Add MS 6783, f. 389v, where Harriot proved that <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>.
 </emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1294" xml:space="preserve">
Est igitur:
<lb/>[<emph style="it">tr: 
Therefore it is true:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1295" xml:space="preserve">
Synthesis
<lb/>[<emph style="it">tr: 
Synthesis
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1296" xml:space="preserve">
Quod fuit propositum.
<lb/>[<emph style="it">tr: 
Which was proposed.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1297" xml:space="preserve">
Vel
<lb/>[<emph style="it">tr: 
Or
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1298" xml:space="preserve">
Sint:
<lb/>[<emph style="it">tr: 
Let ... [be in continued proportion]:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1299" xml:space="preserve">
hoc est:
<lb/>[<emph style="it">tr: 
that is:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1300" xml:space="preserve">
sit:
<lb/>[<emph style="it">tr: 
let:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1301" xml:space="preserve">
Dico quod
<lb/>[<emph style="it">tr: 
I say that
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1302" xml:space="preserve">
hoc est:
<lb/>[<emph style="it">tr: 
that is:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1303" xml:space="preserve">
hoc est:
<lb/>[<emph style="it">tr: 
that is:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1304" xml:space="preserve">
Aliter
<lb/>[<emph style="it">tr: 
Another way
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1305" xml:space="preserve">
Ergo:
<lb/>[<emph style="it">tr: 
Therefore:
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f389v" o="389v" n="778"/>
<div xml:id="echoid-div213" type="page_commentary" level="2" n="213">
<p>
<s xml:id="echoid-s1306" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1306" xml:space="preserve">
In this page Harriot proves the inequality <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>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s1308" xml:space="preserve">
Lemma
<lb/>[<emph style="it">tr: 
Lemma
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1309" xml:space="preserve">
Sit.
<lb/>[<emph style="it">tr: 
Let
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1310" xml:space="preserve">
Et sit:
<lb/>[<emph style="it">tr: 
And let:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1311" 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-s1312" xml:space="preserve">
ponatur.
<lb/>[<emph style="it">tr: 
suppose it is true.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1313" xml:space="preserve">
Tum:
<lb/>[<emph style="it">tr: 
Then:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1314" xml:space="preserve">
Hoc est:
<lb/>[<emph style="it">tr: 
That is:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1315" xml:space="preserve">
Ergo:
<lb/>[<emph style="it">tr: 
Therefore:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1316" xml:space="preserve">
Sit
<lb/>[<emph style="it">tr: 
Let
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1317" xml:space="preserve">
Ergo:
<lb/>[<emph style="it">tr: 
Therefore:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1318" xml:space="preserve">
Hoc est:
<lb/>[<emph style="it">tr: 
That is:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1319" xml:space="preserve">
Ergo:
<lb/>[<emph style="it">tr: 
Therefore:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1320" xml:space="preserve">
Ergo:
<lb/>[<emph style="it">tr: 
Therefore:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1321" xml:space="preserve">
Sit
<lb/>[<emph style="it">tr: 
Let
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1322" xml:space="preserve">
Dico quod:
<lb/>[<emph style="it">tr: 
I say that:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1323" xml:space="preserve">
Hoc est:
<lb/>[<emph style="it">tr: 
That is:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1324" xml:space="preserve">
vel:
<lb/>[<emph style="it">tr: 
or:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1325" xml:space="preserve">
Vel:
<lb/>[<emph style="it">tr: 
Or:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1326" xml:space="preserve">
Si sint tres continue proportiona-<lb/>
les inæquales aggregatum <lb/>
ex maxima et minima, maius <lb/>
est bis media.
<lb/>[<emph style="it">tr: 
If there are three continued proportionals,
</emph>]<lb/>the sum of the greatest and the least is greater than twice the mean.
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1327" xml:space="preserve">
Aliter.
<lb/>[<emph style="it">tr: 
Another way.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1328" xml:space="preserve">
Ergo.
<lb/>[<emph style="it">tr: 
Therefore.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f390" o="390" n="779"/>
<pb file="add_6783_f390v" o="390v" n="780"/>
<pb file="add_6783_f391" o="391" n="781"/>
<pb file="add_6783_f391v" o="391v" n="782"/>
<pb file="add_6783_f392" o="392" n="783"/>
<pb file="add_6783_f392v" o="392v" n="784"/>
<div xml:id="echoid-div214" type="page_commentary" level="2" n="214">
<p>
<s xml:id="echoid-s1329" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1329" xml:space="preserve">
This folio is a summary of Harriot's findings in the six consecutively numbered sheets:
Add MS 6783, f. 334v, 333v, Add MS 6782, f. 55, 56, Add MS 6783, f. 386v, 387v,
that the cube of any binome is a binome of the same kind. <lb/>
Harriot demonstrates this again for a general binome <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>c</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>d</mi><mi>f</mi></mrow></msqrt></mstyle></math>,
and then again, more simply, at the bottom of the page for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>c</mi></mrow></msqrt></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head212" xml:space="preserve" xml:lang="lat">
Quod cubus binomij, est <emph style="st">cubus</emph> binomi eiusdem ordinis, <lb/>
et Differentia quadratum est cubus; cuius radix cubum <lb/>
est differentia <lb/>
quadratorum radicis <lb/>
binomij
<lb/>[<emph style="it">tr: 
That a cube of a binome is a binome of the same order, <lb/>
and the difference of the squares is a cube,
whose cube root is the difference of the roots of the squares in the binome.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1331" xml:space="preserve">
D[iffe]r[enti]a. quadratorum est cubus.
<lb/>[<emph style="it">tr: 
The difference of the squares is a cube.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1332" xml:space="preserve">
Aliter:
<lb/>[<emph style="it">tr: 
Another way:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1333" xml:space="preserve">
D[iffe]r[enti]a. quad[rata].
<lb/>[<emph style="it">tr: 
The difference of the squares.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f393" o="393" n="785"/>
<pb file="add_6783_f393v" o="393v" n="786"/>
<div xml:id="echoid-div215" type="page_commentary" level="2" n="215">
<p>
<s xml:id="echoid-s1334" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1334" xml:space="preserve">
The rule for the square root of a binome of the first kind (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi><mo>-</mo><mi>c</mi><mi>c</mi></mrow></msqrt></mstyle></math>) is stated
and then checked by multiplication.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6783_f394" o="394" n="787"/>
<pb file="add_6783_f394v" o="394v" n="788"/>
<div xml:id="echoid-div216" type="page_commentary" level="2" n="216">
<p>
<s xml:id="echoid-s1336" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1336" xml:space="preserve">
The rule for the square root of a binome of the sixth kind (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>4</mn><mi>b</mi><mi>c</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mn>4</mn><mi>b</mi><mi>c</mi><mo>-</mo><mn>4</mn><mi>d</mi><mi>f</mi></mrow></msqrt></mstyle></math>),
checked by multiplication. A rule for the sixth binome is easily adapted to all other binomes,
which can be regarded as particular cases of it.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s1338" xml:space="preserve">
species universalis
<lb/>[<emph style="it">tr: 
general case
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1339" xml:space="preserve">
exam: per numeros
<lb/>[<emph style="it">tr: 
examined in numbers
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f395" o="395" n="789"/>
<pb file="add_6783_f395v" o="395v" n="790"/>
<div xml:id="echoid-div217" type="page_commentary" level="2" n="217">
<p>
<s xml:id="echoid-s1340" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1340" xml:space="preserve">
This folio provides a canon, or rule, for finding the cube root of a quantity <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>+</mo><mi>f</mi></mstyle></math>. <lb/>
Using the fundamental property established in the first line,
the first denominator of the canon can be rewritten as the cube root of <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>6</mn><mi>b</mi><mi>c</mi><mi>c</mi><mo>+</mo><mn>3</mn><mo maxsize="1">(</mo><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>b</mi><mi>b</mi><mo>-</mo><mi>c</mi><mi>c</mi><mo maxsize="1">)</mo><mo>+</mo><mo maxsize="1">(</mo><mi>b</mi><mo>-</mo><mi>c</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>b</mi><mi>b</mi><mo>-</mo><mi>c</mi><mi>c</mi><mo maxsize="1">)</mo><mo maxsize="1">)</mo></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mn>2</mn><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>6</mn><mi>b</mi><mi>c</mi><mi>c</mi><mo>+</mo><mn>6</mn><mi>b</mi><mo maxsize="1">(</mo><mi>b</mi><mi>b</mi><mo>-</mo><mi>c</mi><mi>c</mi><mo maxsize="1">)</mo></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mn>8</mn><mi>b</mi><mi>b</mi><mi>b</mi></mstyle></math>. <lb/>
Similarly, the second denominator yields <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>8</mn><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>.
Thus taking cube roots and dividing by 2 yields <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>, as required. <lb/>
A numerical example is worked at the bottom of the page.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head213" xml:space="preserve" xml:lang="lat">
De extractione radicis cubicæ <lb/>
e solido binomio. (χαθολιχῶς.)
<lb/>[<emph style="it">tr: 
On the extraction of cube roots of binomes (general)
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1342" xml:space="preserve">
Fundamentum
<lb/>[<emph style="it">tr: 
Fundamental property
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1343" xml:space="preserve">
Sit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>&gt;</mo><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
Suppose: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>&gt;</mo><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1344" xml:space="preserve">
ergo; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>&gt;</mo><mi>f</mi></mstyle></math>.
<lb/>[<emph style="it">tr: 
therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>&gt;</mo><mi>f</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1345" xml:space="preserve">
per lemma.
<lb/>[<emph style="it">tr: 
by the lemma
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1346" xml:space="preserve">
Canon.
<lb/>[<emph style="it">tr: 
Rule.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1347" xml:space="preserve">
Illustratio Canonis per numeros.
<lb/>[<emph style="it">tr: 
An illustration of the rule in numbers.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f396" o="396" n="791"/>
<pb file="add_6783_f396v" o="396v" n="792"/>
<div xml:id="echoid-div218" type="page_commentary" level="2" n="218">
<p>
<s xml:id="echoid-s1348" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1348" xml:space="preserve">
On this folio Harriot explains how to find the cube root of a quantity of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>f</mi><mi>f</mi><mi>f</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>g</mi><mi>g</mi><mi>g</mi></mrow></msqrt></mstyle></math>,
where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>f</mi><mi>f</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>g</mi><mi>g</mi></mstyle></math> take the particular form shown, so that their difference is the cube of <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>.
Harriot derives a cubic equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, enabling him to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi></mrow></msqrt></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>c</mi></mrow></msqrt></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head214" xml:space="preserve" xml:lang="lat">
De extractione radicis cubicæ e binomio solido, cuius <lb/>
quadratorum differentia est numerus cubicus. <lb/>
Quale binomium semper invenitur in resolutione <lb/>
æquationum (e.5. et e.6.)
<lb/>[<emph style="it">tr: 
On the extraction of cube roots of a solid binome,
in which the difference of the squares is a cube number.
Which binomes are always found in the resolution of equations (e.5 and e.6).
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1350" xml:space="preserve">
Differentia <lb/>
quadratorum
<lb/>[<emph style="it">tr: 
Difference of the squares
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1351" xml:space="preserve">
Omnia probantur <lb/>
per numeros <lb/>
et <lb/>
recte se habeat.
<lb/>[<emph style="it">tr: 
All verified in numbers and found to be correct.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1352" xml:space="preserve">
Ergo pro Canone:
<lb/>[<emph style="it">tr: 
Therefore by the rule:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1353" xml:space="preserve">
Sed etiam
<lb/>[<emph style="it">tr: 
But also
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1354" xml:space="preserve">
Ergo datur:
<lb/>[<emph style="it">tr: 
Therefore we have:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1355" xml:space="preserve">
Differentia <lb/>
quadratorum
<lb/>[<emph style="it">tr: 
Difference of the squares
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1356" xml:space="preserve">
Ex istis Fundamentis, oporte invenire (e) via resolvendi <lb/>
generali. alias resolvendum esset aliud binomium solidum <lb/>
quod est petere principium.
<lb/>[<emph style="it">tr: 
From these foundations, one may find (e) the general method of solution.
otherwise there must be resolved another solid binome, which is to ask for the first one.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f397" o="397" n="793"/>
<pb file="add_6783_f397v" o="397v" n="794"/>
<pb file="add_6783_f398" o="398" n="795"/>
<pb file="add_6783_f398v" o="398v" n="796"/>
<pb file="add_6783_f399" o="399" n="797"/>
<pb file="add_6783_f399v" o="399v" n="798"/>
<pb file="add_6783_f400" o="400" n="799"/>
<pb file="add_6783_f400v" o="400v" n="800"/>
<pb file="add_6783_f401" o="401" n="801"/>
<pb file="add_6783_f401v" o="401v" n="802"/>
<div xml:id="echoid-div219" type="page_commentary" level="2" n="219">
<p>
<s xml:id="echoid-s1357" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1357" xml:space="preserve">
The frequency analysis on this page matches that on Add MS 6783, f. 410.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head215" xml:space="preserve" xml:lang="lat">
e. De resolutione æquationum per reductionum
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1359" xml:space="preserve">
bonum. bene. est. virtus libere <lb/>
omnibus <lb/>
omnibus est virtus non rara bibere <lb/>
esse bonum valat at bibere
</s>
</p>
<pb file="add_6783_f402" o="402" n="803"/>
<head xml:id="echoid-head216" xml:space="preserve" xml:lang="lat">
e.7. Aliter. casus. 1.
<lb/>[<emph style="it">tr: 
Another way. Case 1.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1360" xml:space="preserve">
melius
<lb/>[<emph style="it">tr: 
better
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f402v" o="402v" n="804"/>
<pb file="add_6783_f403" o="403" n="805"/>
<div xml:id="echoid-div220" type="page_commentary" level="2" n="220">
<p>
<s xml:id="echoid-s1361" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1361" xml:space="preserve">
See also Add MS 6783, f. 401v, f. 410.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s1363" xml:space="preserve">
bene valere eset bonum
</s>
</p>
<pb file="add_6783_f403v" o="403v" n="806"/>
<pb file="add_6783_f404" o="404" n="807"/>
<div xml:id="echoid-div221" type="page_commentary" level="2" n="221">
<p>
<s xml:id="echoid-s1364" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1364" xml:space="preserve">
Rough work for Add MS 6783, f. 98 or Add MS 6783, f. 406v.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6783_f404v" o="404v" n="808"/>
<pb file="add_6783_f405" o="405" n="809"/>
<pb file="add_6783_f405v" o="405v" n="810"/>
<pb file="add_6783_f406" o="406" n="811"/>
<pb file="add_6783_f406v" o="406v" n="812"/>
<div xml:id="echoid-div222" type="page_commentary" level="2" n="222">
<p>
<s xml:id="echoid-s1366" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1366" xml:space="preserve">
A draft or alternative version for Add MS 6783, f. 98.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head217" xml:space="preserve" xml:lang="lat">
e.1.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1368" xml:space="preserve">
Apparatus, ad genera, species et differentius <lb/>
æquationum adventiturum.
<lb/>[<emph style="it">tr: 
Preparation, on the degrees, types, and cases of the equations in question.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1369" xml:space="preserve">
Collectio <lb/>
et <lb/>
summaria <lb/>
numeratio.
<lb/>[<emph style="it">tr: 
A gathering and summary of the enumeration.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f407" o="407" n="813"/>
<head xml:id="echoid-head218" xml:space="preserve">
e.6.
</head>
<pb file="add_6783_f407v" o="407v" n="814"/>
<pb file="add_6783_f408" o="408" n="815"/>
<pb file="add_6783_f408v" o="408v" n="816"/>
<pb file="add_6783_f409" o="409" n="817"/>
<pb file="add_6783_f409v" o="409v" n="818"/>
<div xml:id="echoid-div223" type="page_commentary" level="2" n="223">
<p>
<s xml:id="echoid-s1370" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1370" xml:space="preserve">
Rough work for Add MS 6783, f. 98 or Add MS 6783, f. 406v.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head219" xml:space="preserve" xml:lang="lat">
e.1.) De resolutione æquationum per reductione
<lb/>[<emph style="it">tr: 
On the solution of equations by reduction
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s1372" xml:space="preserve">
Apparatus, ad genera, species et differentius <lb/>
æquationum adventiturum.
<lb/>[<emph style="it">tr: 
Preparation, on the degrees, types, and cases of the equations in question.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f410" o="410" n="819"/>
<div xml:id="echoid-div224" type="page_commentary" level="2" n="224">
<p>
<s xml:id="echoid-s1373" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1373" xml:space="preserve">
At the bottom of this page is what appears to be a frequency analysis of a phrase containing 35 letters.
See also Add MS 6783, f. 401v.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s1375" xml:space="preserve">
Hexameter <lb/>
ex his <emph style="super">sequentibus</emph> litteribus compositus <lb/>
[???] <lb/>
explicat quid sunt <lb/>
includæ in [???] apparentes.
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1376" xml:space="preserve">
litteræ specie sunt, 13 <lb/>
numero. 35
<lb/>[<emph style="it">tr: 
Types of letter, 13; number 35.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f410v" o="410v" n="820"/>
<pb file="add_6783_f411" o="411" n="821"/>
<pb file="add_6783_f411v" o="411v" n="822"/>
<pb file="add_6783_f412" o="412" n="823"/>
<pb file="add_6783_f412v" o="412v" n="824"/>
<pb file="add_6783_f413" o="413" n="825"/>
<pb file="add_6783_f413v" o="413v" n="826"/>
<pb file="add_6783_f414" o="414" n="827"/>
<pb file="add_6783_f414v" o="414v" n="828"/>
<pb file="add_6783_f415" o="415" n="829"/>
<pb file="add_6783_f415v" o="415v" n="830"/>
<pb file="add_6783_f416" o="416" n="831"/>
<pb file="add_6783_f416v" o="416v" n="832"/>
<pb file="add_6783_f417" o="417" n="833"/>
<pb file="add_6783_f417v" o="417v" n="834"/>
<pb file="add_6783_f418" o="418" n="835"/>
<pb file="add_6783_f418v" o="418v" n="836"/>
<pb file="add_6783_f419" o="419" n="837"/>
<pb file="add_6783_f419v" o="419v" n="838"/>
<pb file="add_6783_f420" o="420" n="839"/>
<pb file="add_6783_f420v" o="420v" n="840"/>
<pb file="add_6783_f421" o="421" n="841"/>
<div xml:id="echoid-div225" type="page_commentary" level="2" n="225">
<p>
<s xml:id="echoid-s1377" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1377" xml:space="preserve">
Some quartic equations and their roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6783_f421v" o="421v" n="842"/>
<div xml:id="echoid-div226" type="page_commentary" level="2" n="226">
<p>
<s xml:id="echoid-s1379" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1379" xml:space="preserve">
Some quartic equations and their roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6783_f422" o="422" n="843"/>
<div xml:id="echoid-div227" type="page_commentary" level="2" n="227">
<p>
<s xml:id="echoid-s1381" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1381" xml:space="preserve">
Some cubic equations and their roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s1383" xml:space="preserve">
eadem <lb/>
exempla <lb/>
pro coniug-<lb/>
ata
<lb/>[<emph style="it">tr: 
the same example for conjugates
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1384" xml:space="preserve">
vide. α.1
<lb/>[<emph style="it">tr: 
see α.1
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1385" xml:space="preserve">
nulla reciproca in his.
<lb/>[<emph style="it">tr: 
no reciprocals in these.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1386" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>&gt;</mo><mi>a</mi></mstyle></math> super.
<lb/>[<emph style="it">tr: 
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>&gt;</mo><mi>a</mi></mstyle></math> above.
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f422v" o="422v" n="844"/>
<pb file="add_6783_f423" o="423" n="845"/>
<div xml:id="echoid-div228" type="page_commentary" level="2" n="228">
<p>
<s xml:id="echoid-s1387" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1387" xml:space="preserve">
Some cubic equations and their roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s1389" xml:space="preserve">
casus
<lb/>[<emph style="it">tr: 
case
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1390" xml:space="preserve">
reciproca
<lb/>[<emph style="it">tr: 
reciprocal
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1391" xml:space="preserve">
ut infra <lb/>
melius
<lb/>[<emph style="it">tr: 
as below, better
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1392" xml:space="preserve">
casus
<lb/>[<emph style="it">tr: 
case
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1393" xml:space="preserve">
reciproca
<lb/>[<emph style="it">tr: 
reciprocal
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1394" xml:space="preserve">
ut supra
<lb/>[<emph style="it">tr: 
as above
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f423v" o="423v" n="846"/>
<pb file="add_6783_f424" o="424" n="847"/>
<div xml:id="echoid-div229" type="page_commentary" level="2" n="229">
<p>
<s xml:id="echoid-s1395" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1395" xml:space="preserve">
Some cubic equations and their roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s1397" xml:space="preserve">
reciproca
<lb/>[<emph style="it">tr: 
reciprocal
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s1398" xml:space="preserve">
In minimis numeris
<lb/>[<emph style="it">tr: 
In least numbers
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1399" xml:space="preserve">
Vietæ exempla
<lb/>[<emph style="it">tr: 
Viète's examples
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1400" xml:space="preserve">
Triplæ q: (e triente) coeffic long:
<lb/>[<emph style="it">tr: 
Triple the square (of a third) of the coefficient of the square term.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s1401" xml:space="preserve">
reciproca
<lb/>[<emph style="it">tr: 
reciprocal
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f424v" o="424v" n="848"/>
<pb file="add_6783_f425" o="425" n="849"/>
<pb file="add_6783_f425v" o="425v" n="850"/>
<div xml:id="echoid-div230" type="page_commentary" level="2" n="230">
<p>
<s xml:id="echoid-s1402" xml:space="preserve">[<emph style="it">Note: 
<p>
<s xml:id="echoid-s1402" xml:space="preserve">
Some cubic equations and their roots.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s1404" xml:space="preserve">
reciproca
<lb/>[<emph style="it">tr: 
reciprocal
</emph>]<lb/>
</s>
</p>
<pb file="add_6783_f426" o="426" n="851"/>
<pb file="add_6783_f426v" o="426v" n="852"/>
</div>
</text>
</echo>