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diff texts/XML/echo/en/Harriot_Add_MS_6782_HSPGZ0AE.xml @ 6:22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
| author | casties |
|---|---|
| date | Fri, 07 Dec 2012 17:05:22 +0100 |
| parents | |
| children | 568e026bb6d6 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/texts/XML/echo/en/Harriot_Add_MS_6782_HSPGZ0AE.xml Fri Dec 07 17:05:22 2012 +0100 @@ -0,0 +1,17713 @@ +<?xml version="1.0" encoding="utf-8"?><echo xmlns="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:de="http://www.mpiwg-berlin.mpg.de/ns/de/1.0/" xmlns:dcterms="http://purl.org/dc/terms" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:xhtml="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.0RC"> +<metadata> +<dcterms:identifier>ECHO:HSPGZ0AE.xml</dcterms:identifier> +<dcterms:creator>Harriot, Thomas</dcterms:creator> +<dcterms:title xml:lang="en">Mss. 6782</dcterms:title> +<dcterms:date xsi:type="dcterms:W3CDTF">o. J.</dcterms:date> +<dcterms:language xsi:type="dcterms:ISO639-3">eng</dcterms:language> +<dcterms:rights>CC-BY-SA</dcterms:rights> +<dcterms:license xlink:href="http://creativecommons.org/licenses/by-sa/3.0/">CC-BY-SA</dcterms:license> +<dcterms:rightsHolder xlink:href="http://www.mpiwg-berlin.mpg.de">Max Planck Institute for the History of Science, Library</dcterms:rightsHolder> +<echodir>/permanent/library/HSPGZ0AE</echodir> +<log>Automatically generated by bare_xml.py on Tue Nov 15 14:20:53 2011</log> +</metadata> + +<text xml:lang="eng" type="free"> +<div xml:id="echoid-div1" type="bundle" level="1" n="1"> +<pb file="add_6782_f001" o="1" n="1"/> +<div xml:id="echoid-div1" type="page_commentary" level="2" n="1"> +<p> +<s xml:id="echoid-s1" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1" xml:space="preserve"> +Lists of powers of two up to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mn>2</mn><mrow><mn>2</mn><mn>9</mn></mrow></msup></mrow></mstyle></math>, in standard denary (right) and in octonary (left). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head1" xml:space="preserve" xml:lang="lat"> +Octonaria. Denaria. +<lb/>[<emph style="it">tr: +Octonary. Denary +</emph>]<lb/> +</head> +<pb file="add_6782_f001v" o="1v" n="2"/> +<pb file="add_6782_f002" o="2" n="3"/> +<pb file="add_6782_f002v" o="2v" n="4"/> +<pb file="add_6782_f003" o="3" n="5"/> +<pb file="add_6782_f003v" o="3v" n="6"/> +<pb file="add_6782_f004" o="4" n="7"/> +<pb file="add_6782_f004v" o="4v" n="8"/> +<pb file="add_6782_f005" o="5" n="9"/> +<pb file="add_6782_f005v" o="5v" n="10"/> +<pb file="add_6782_f006" o="6" n="11"/> +<pb file="add_6782_f006v" o="6v" n="12"/> +<pb file="add_6782_f007" o="7" n="13"/> +<pb file="add_6782_f007v" o="7v" n="14"/> +<pb file="add_6782_f008" o="8" n="15"/> +<pb file="add_6782_f008v" o="8v" n="16"/> +<pb file="add_6782_f009" o="9" n="17"/> +<pb file="add_6782_f009v" o="9v" n="18"/> +<pb file="add_6782_f010" o="10" n="19"/> +<pb file="add_6782_f010v" o="10v" n="20"/> +<pb file="add_6782_f011" o="11" n="21"/> +<pb file="add_6782_f011v" o="11v" n="22"/> +<pb file="add_6782_f012" o="12" n="23"/> +<pb file="add_6782_f012v" o="12v" n="24"/> +<pb file="add_6782_f013" o="13" n="25"/> +<pb file="add_6782_f013v" o="13v" n="26"/> +<pb file="add_6782_f014" o="14" n="27"/> +<pb file="add_6782_f014v" o="14v" n="28"/> +<pb file="add_6782_f015" o="15" n="29"/> +<pb file="add_6782_f015v" o="15v" n="30"/> +<pb file="add_6782_f016" o="16" n="31"/> +<pb file="add_6782_f016v" o="16v" n="32"/> +<pb file="add_6782_f017" o="17" n="33"/> +<pb file="add_6782_f017v" o="17v" n="34"/> +<pb file="add_6782_f018" o="18" n="35"/> +<pb file="add_6782_f018v" o="18v" n="36"/> +<pb file="add_6782_f019" o="19" n="37"/> +<pb file="add_6782_f019v" o="19v" n="38"/> +<pb file="add_6782_f020" o="20" n="39"/> +<pb file="add_6782_f020v" o="20v" n="40"/> +<pb file="add_6782_f021" o="21" n="41"/> +<pb file="add_6782_f021v" o="21v" n="42"/> +<pb file="add_6782_f022" o="22" n="43"/> +<pb file="add_6782_f022v" o="22v" n="44"/> +<pb file="add_6782_f023" o="23" n="45"/> +<pb file="add_6782_f023v" o="23v" n="46"/> +<pb file="add_6782_f024" o="24" n="47"/> +<pb file="add_6782_f024v" o="24v" n="48"/> +<pb file="add_6782_f025" o="25" n="49"/> +<pb file="add_6782_f025v" o="25v" n="50"/> +<pb file="add_6782_f026" o="26" n="51"/> +<pb file="add_6782_f026v" o="26v" n="52"/> +<pb file="add_6782_f027" o="27" n="53"/> +<div xml:id="echoid-div2" type="page_commentary" level="2" n="2"> +<p> +<s xml:id="echoid-s3" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s3" xml:space="preserve"> +A word square based on the words HENRICUS PRINCEPS FECIT (Prince Henry made it). <lb/> +The number 184,756 appears four times in the bottom right. +In each quarter of the square, there are 184,756 ways of reading HENRICUS PRINCEPS FECIT, +starting from the centre and ending at a corner. Thus there are 739,024 ways in total. +This number may have been written on the page but has now disappeared (see Add MS 6782, f. 28 for a similar calculation). +For the calculations leading to 184,756 see Add MS 6782, f. 57 and f. 58. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f027v" o="27v" n="54"/> +<pb file="add_6782_f028" o="28" n="55"/> +<div xml:id="echoid-div3" type="page_commentary" level="2" n="3"> +<p> +<s xml:id="echoid-s5" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s5" xml:space="preserve"> +A word square based on the words SILO PRINCEPS FECIT (Prince Henry made it). <lb/> +A <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>5</mn><mo>×</mo><mn>1</mn><mn>9</mn></mstyle></math> rectangular version of this arrangement, carved in stone, is to be found in the church of +San Juan Apostol y Evangelista in Santianes de Pravia, northern Spain, +commemorating Silo, king of Asturias (774 to 783). <lb/> +The number 12,780 appears four times in the bottom right hand corner of the page. +In each quarter of the square, there are 12,780 ways of reading SILO PRINCEPS FECIT, +starting from the centre and ending at a corner. Thus there are 51,480 ways in total. +For the calculations leading to 12,780 see Add MS 6782, f. 57 and f. 58. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f028v" o="28v" n="56"/> +<pb file="add_6782_f029" o="29" n="57"/> +<div xml:id="echoid-div4" type="page_commentary" level="2" n="4"> +<p> +<s xml:id="echoid-s7" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s7" xml:space="preserve"> +This folios shows calculations for solving the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>z</mi><mo>+</mo><mn>3</mn><mi>a</mi><mo>=</mo><mn>2</mn><mn>7</mn><mn>6</mn></mstyle></math> +(in modern notation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>3</mn><mi>x</mi><mo>=</mo><mn>2</mn><mn>7</mn><mn>6</mn></mstyle></math>). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f029v" o="29v" n="58"/> +<div xml:id="echoid-div5" type="page_commentary" level="2" n="5"> +<p> +<s xml:id="echoid-s9" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s9" xml:space="preserve"> +Triangular numbers arranged as dot patterns. <lb/> +The arrangement of Pascal's triangle on the right shows how each row is obtained +by summing two copies of the preceding row. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f030" o="30" n="59"/> +<div xml:id="echoid-div6" type="page_commentary" level="2" n="6"> +<p> +<s xml:id="echoid-s11" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s11" xml:space="preserve"> +The upper left quarter of the page contains two versions of Pascal's triangle, in different layouts. <lb/> +The upper right quarter shows the entries of the triangle +generated by successive multiplications of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo maxsize="1">)</mo></mstyle></math>. <lb/> +The lower right quarter shows the entries of the triangle +generated by successive multiplications of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo maxsize="1">)</mo></mstyle></math>. <lb/> +The lower left quarter demonstrates, following from the multiplications on the right, +that the entries in each row sum to a power of 2. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head2" xml:space="preserve" xml:lang="lat"> +De combinationibus +<lb/>[<emph style="it">tr: +On combinations +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s13" xml:space="preserve"> +To serve ones turne. <lb/> +</s> +<s xml:id="echoid-s14" xml:space="preserve"> +To do one a sound turne. +</s> +</p> +<pb file="add_6782_f030v" o="30v" n="60"/> +<div xml:id="echoid-div7" type="page_commentary" level="2" n="7"> +<p> +<s xml:id="echoid-s15" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s15" xml:space="preserve"> +This page includes numerals from 1 to 9 written with a medieval form for '4', +and also with characters composed only of straight lines. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f031" o="31" n="61"/> +<div xml:id="echoid-div8" type="page_commentary" level="2" n="8"> +<p> +<s xml:id="echoid-s17" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s17" xml:space="preserve"> +The reference to 'Saxton's great map' is to Christopher Saxton's county maps for England and Wales, +published from 1579 onwards. <lb/> +Units of measurement: +a pase or pace (from Latin passus) was the length of a double stride, about 5 feet or 1.5 metres. +Thus one square pase was 25 square feet. +The Roman mile was 1000 pases. <lb/> +For some rough working for this page see Add MS 6788, f. 547v. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head3" xml:space="preserve"> +An æstimable reckoning how many persons <lb/> +may inhabit the whole world. +</head> +<p> +<s xml:id="echoid-s19" xml:space="preserve"> +Supositions. +</s> +<lb/> +<s xml:id="echoid-s20" xml:space="preserve"> +[1]. The semiperimeter of a circle. 31,415,926 +</s> +<lb/> +<s xml:id="echoid-s21" xml:space="preserve"> +[2]. The semidiameter. 10,000,000 +</s> +</p> +<p> +<s xml:id="echoid-s22" xml:space="preserve"> +The compasse of the earth <lb/> +after the rate of 60 miles <lb/> +to a degree. 21,600 miles +</s> +</p> +<p> +<s xml:id="echoid-s23" xml:space="preserve"> +Ergo: The halfe compasse. 10,300 miles +<sc> +The figure 10,300 is a copying error for 10,800. +The correct figure has been used in the subsequent calculations +</sc> +</s> +<lb/> +<s xml:id="echoid-s24" xml:space="preserve"> +The semidiameter of the earth. 3,437 miles. 747 pases. +</s> +<lb/> +<s xml:id="echoid-s25" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mo>,</mo><mn>4</mn><mn>3</mn><mn>7</mn><mo>,</mo><mn>7</mn><mn>4</mn><mn>7</mn><mo>×</mo><mn>1</mn><mn>0</mn><mo>,</mo><mn>3</mn><mn>0</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>3</mn><mn>5</mn><mo>,</mo><mn>4</mn><mn>0</mn><mn>8</mn><mo>,</mo><mn>7</mn><mn>9</mn><mn>4</mn><mo>,</mo><mn>1</mn><mn>0</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>=</mo></mstyle></math> <lb/> +plano circuli. <lb/> +quod <reg norm="aequatur" type="abbr">æquat</reg> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> +<reg norm="superficies" type="abbr">superf</reg>: <lb/> +terræ et aquæ. +<lb/>[<emph style="it">tr: +a plane circle which equals <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> the surface of land and water. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s26" xml:space="preserve"> +Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mn>5</mn><mo>,</mo><mn>4</mn><mn>0</mn><mn>8</mn><mo>,</mo><mn>7</mn><mn>9</mn><mn>4</mn><mo>,</mo><mn>1</mn><mn>0</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>×</mo><mn>4</mn><mo>=</mo></mstyle></math> <lb/> +141,635,176,400,000 +<reg norm="superficies" type="abbr">superf:</reg> maris et terræ. +<lb/>[<emph style="it">tr: +The surface of sea and land. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s27" xml:space="preserve"> +70,817,588,200,000 = +<reg norm="superficies" type="abbr">superf:</reg> Terræ: vel maris. +<lb/>[<emph style="it">tr: +The surface of land or sea. +</emph>]<lb/> +</s> +</p> +<p> +<s xml:id="echoid-s28" xml:space="preserve"> +49,987 miles square in England +(<foreign xml:lang="lat">ut aliby</foreign> by Saxtons <lb/> +great map) after the rate of 60 miles to a degree <lb/> +including rivers & all wastes. +<sc> +'aliby' is a copying error for 'alibi' (see Add MS 6788, f. 547v). <lb/> +</sc> +</s> +<lb/> +<s xml:id="echoid-s29" xml:space="preserve"> +It lacketh but 13 miles of 50,000. +</s> +</p> +<p> +<s xml:id="echoid-s30" xml:space="preserve"> +50,000 miles. 5,000,000 persons. supposed. <lb/> +1 mile. 100 persons. 70,817,588 miles. 7,081,758,800. persons in the earth. +</s> +</p> +<p> + <s xml:id="echoid-s31" xml:space="preserve">[<emph style="it">Note: +This subcalculation gives 10,000 square pases per person, +converted next to 250,000 square feet, then to 5 and 8/11 acres. +</emph>]</s><lb/> +<s xml:id="echoid-s32" xml:space="preserve"> +1,000,000 pp. 100 persons. <lb/> +10000 pp. 1 person. <lb/> +10000 [square pases] = <lb/> +250,000 [square feet] = <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mfrac><mrow><mn>8</mn></mrow><mrow><mn>1</mn><mn>1</mn></mrow></mfrac></mstyle></math> acres. +(note rivers & waste included <lb/> +as above.) +</s> +</p> +<p> +<s xml:id="echoid-s33" xml:space="preserve"> +6 men may stand in a pase square. <lb/> +</s> +<s xml:id="echoid-s34" xml:space="preserve"> +Therefore 6,000,000 in one mile square. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mo>,</mo><mn>0</mn><mn>8</mn><mn>1</mn><mo>,</mo><mn>5</mn><mn>8</mn><mo>,</mo><mn>8</mn><mn>0</mn><mn>0</mn><mo>×</mo><mn>6</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>×</mo><mn>4</mn><mn>2</mn><mo>,</mo><mn>4</mn><mn>9</mn><mn>0</mn><mo>,</mo><mn>5</mn><mn>5</mn><mn>2</mn><mo>,</mo><mn>8</mn><mn>0</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>,</mo><mn>0</mn></mstyle></math>. <lb/> +</s> +<s xml:id="echoid-s35" xml:space="preserve"> +The number of persons yt may stand on ye earth. +</s> +</p> +<pb file="add_6782_f031v" o="31v" n="62"/> +<div xml:id="echoid-div9" type="page_commentary" level="2" n="9"> +<p> +<s xml:id="echoid-s36" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s36" xml:space="preserve"> +For some of the calculations behind this page see Add MS 6788, f. 536, f. 537, f. 541. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head4" xml:space="preserve"> +The issue from one man & one woman in 240 yeares may be <lb/> +more then can inhabit the whole earth. +</head> +<p> +<s xml:id="echoid-s38" xml:space="preserve"> +Supposing. +</s> +<lb/> +<s xml:id="echoid-s39" xml:space="preserve"> +1. That the first man & woman have a child every yeare <lb/> +one yeare male & in other <emph style="super">yeare</emph> female +</s> +</p> +<p> +<s xml:id="echoid-s40" xml:space="preserve"> +2. That the children when they are 20 yeares old & upward <lb/> +do also every yeare beget a child one yeare male & an <lb/> +other yeare female +</s> +</p> +<p> +<s xml:id="echoid-s41" xml:space="preserve"> +3. That all are living at the end of 240 yeares. <lb/> +</s> +<s xml:id="echoid-s42" xml:space="preserve"> +The number of <lb/> +males, 5,034,303,437. <lb/> +females, 5,034,303,437. <lb/> +persons, 10,068,606,874. <lb/> +(recconed in <foreign xml:lang="lat">charta</foreign> c) +[<emph style="it">Note: +Sheet c) is almost certainly Add MS 6788, f. 537, +where the same suppositions appear and this calculation is carried out, +but unfortunately the lettering of that page is obscured in the binding. + </emph>]<lb/> +</s> +</p> +<p> +<s xml:id="echoid-s43" xml:space="preserve"> +That in 400 yeares upon the former suppositions <lb/> +there would be more men then can stand on the face <lb/> +of the whole <emph style="st">yeare</emph> earth. +</s> +</p> +<p> +<s xml:id="echoid-s44" xml:space="preserve"> +(In <foreign xml:lang="lat">charta</foreign> db) I find that in 340 yeares they will make <lb/> +a number of 14 places. +[<emph style="it">Note: +Sheet db) is probably Add MS 6788, f. 541, but the lettering of that page is unfortunately obscured in the binding. + </emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s45" xml:space="preserve"> +Therefore in 400 yeares they will make a number of <lb/> +16 places which is more then can stand on the face of <lb/> +the earth (<foreign xml:lang="lat">ut versa pagina</foreign>.) +<sc> +'ut versa pagina' (as the other side of the page); see Add MS 6782, f. 31. +</sc> +</s> +</p> +<p> +<s xml:id="echoid-s46" xml:space="preserve"> +How many persons have had being in 6000, yeares <lb/> +and in what roome they may stand. +</s> +</p> +<p> +<s xml:id="echoid-s47" xml:space="preserve"> +Supposing <lb/> +1. That <emph style="st">in 40 yeares</emph> the world when it was replenished <lb/> +to the number of 7,000,000,000. +(<foreign xml:lang="lat">ut versa pagina</foreign>) <lb/> +they were alwayes one time with an other the same number. +<sc> +'ut versa pagina' (as the other side of the page); see Add MS 6782, f. 31. +</sc> +</s> +</p> +<p> +<s xml:id="echoid-s48" xml:space="preserve"> +2. That in <emph style="st">every</emph> 40 yeares there is (one time with an other) as <lb/> +I have proved in an other page (& agreeth with Nombers .26.64 <lb/> +a new generation. +</s> +</p> +<p> +<s xml:id="echoid-s49" xml:space="preserve"> +Therefore in 6000 yeares there are 150 generations. <lb/> +</s> +<s xml:id="echoid-s50" xml:space="preserve"> +Therefore there have been of persons in 6000 yeres <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>×</mo><mn>1</mn><mn>5</mn><mn>0</mn><mo>=</mo><mn>1</mn><mo>,</mo><mn>0</mn><mn>5</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn></mstyle></math>. persons. <lb/> +</s> +<s xml:id="echoid-s51" xml:space="preserve"> +There being 50,000 miles square in England; <emph style="st">therefore</emph> there <lb/> +may stand in England 300,000,000,000. persons. <lb/> +</s> +<s xml:id="echoid-s52" xml:space="preserve"> +The former number is three times greater, & therefore <lb/> +there place of standing must be also 3 times greater then England. +</s> +</p> +<pb file="add_6782_f032" o="32" n="63"/> +<div xml:id="echoid-div10" type="page_commentary" level="2" n="10"> +<p> +<s xml:id="echoid-s53" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s53" xml:space="preserve"> +Another copy of the table shown on Add MS 6782, f. 63. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f032v" o="32v" n="64"/> +<pb file="add_6782_f033" o="33" n="65"/> +<div xml:id="echoid-div11" type="page_commentary" level="2" n="11"> +<p> +<s xml:id="echoid-s55" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s55" xml:space="preserve"> +This folio gives systematic lists of all combinations (without repetition) of the letters +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi><mi>e</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi><mi>e</mi><mi>f</mi></mstyle></math>. +In each case the combinations are listed as single letters, pairs, triples, and so on. +Combinations of the same size are listed alphabetically, thus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>d</mi></mstyle></math>, and so on. +(In the final column the triples are to be read downwards from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>e</mi><mi>f</mi></mstyle></math> +then back up from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>f</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi><mi>f</mi></mstyle></math>.) <lb/> +Numbers written to the right of the columns show the number of combinations in each part of the list. +Totals are given at the bottom. <lb/> +A table mid-left lists numbers of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mn>2</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mn>2</mn><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><mo>-</mo><mn>1</mn></mstyle></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>=</mo><mn>1</mn></mstyle></math>. +The latter are the numbers that appear as totals. <lb/> +A table lower left shows how combinations of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math> may be derived from those for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> +by adding <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> to the end of each of them and then also listing <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> as a singleton. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head5" xml:space="preserve"> +Of combinations +</head> +<pb file="add_6782_f033v" o="33v" n="66"/> +<pb file="add_6782_f034" o="34" n="67"/> +<div xml:id="echoid-div12" type="page_commentary" level="2" n="12"> +<p> +<s xml:id="echoid-s57" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s57" xml:space="preserve"> +This folio gives systematic lists of all combinations (without repetition) of +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi><mi>e</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi><mi>e</mi><mi>f</mi></mstyle></math>. +In each case the combinations are listed as single letters, pairs, triples, nd so on. +Combinations of the same kind are listed alphabetically, thus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>d</mi></mstyle></math>, and so on. +Numbers written to the right of the columns show the number of combinations in each part of the list. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head6" xml:space="preserve"> +Of combinations +</head> +<pb file="add_6782_f034v" o="34v" n="68"/> +<pb file="add_6782_f035" o="35" n="69"/> +<div xml:id="echoid-div13" type="page_commentary" level="2" n="13"> +<p> +<s xml:id="echoid-s59" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s59" xml:space="preserve"> +This folio shows combinations (without repetition) of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>, +with each list constructed from the previous one. +Combinations of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>, for example, are found from combinations of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math> +by adding <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> to each of them, together with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> as a singleton. <lb/> +Numbers to the right of each column show the number of combinations in each part of the list. +Totals are given at the bottom. <lb/> +Harriot uses the word 'complications' for combinations of more than one letter, +and 'simples' for single letters. <lb/> +At the end of the page Harriot reaches the conclusion that if null combinations are counted, +then the total number of combinations will be a power of 2. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head7" xml:space="preserve"> +Of combinations +</head> +<p> +<s xml:id="echoid-s61" xml:space="preserve"> +By this manner of construction & <lb/> +generation of the variety of combinations <lb/> +or complications: <lb/> +these propositions are manifest: +</s> +</p> +<p> +<s xml:id="echoid-s62" xml:space="preserve"> +The nomber of complications with the <lb/> +nomber of there simples: is double to the <lb/> +nomber of complications with there simples, <lb/> +of the next praecedent order: & one more. +</s> +</p> +<p> +<s xml:id="echoid-s63" xml:space="preserve"> +In any order of complications: +</s> +</p> +<p> +<s xml:id="echoid-s64" xml:space="preserve"> +The nomber of <lb/> +Bynaryes Ternaryes Quaternaries &c. <lb/> +is æquall to the nomber, in the precedent order, of: <lb/> +Binaryes & Simples. +Ternary<emph style="super">es</emph>s & Binaryes. +Quaternaryes & Ternaryes. & c. +</s> +</p> +<p> +<s xml:id="echoid-s65" xml:space="preserve"> +Hereby is <emph style="st">[???]</emph> also manifest, the reason & order <lb/> +of the construction of the table of combinations by nombers <lb/> +which <emph style="st">followeth</emph> <emph style="super">is set downe</emph> +in an other paper. +</s> +</p> +<p> +<s xml:id="echoid-s66" xml:space="preserve"> +In some cases rationall or negative <lb/> +is <emph style="st">[???]</emph> added: as of the species <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. question may be made whether an other <lb/> +thing hath acte upon. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. or both <lb/> +that is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. or neither. +</s> +</p> +<p> +<s xml:id="echoid-s67" xml:space="preserve"> +And such negatives may be understode <lb/> +of all the rest. +</s> +<s xml:id="echoid-s68" xml:space="preserve"> +And thus the sommes of <lb/> +every order wilbe one more; and there <lb/> +progression wilbe. 2. 4. 8. 16. 32. &c. +</s> +</p> +<p> +<s xml:id="echoid-s69" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, or not. <lb/> +</s> +<s xml:id="echoid-s70" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. or not. <lb/> +</s> +<s xml:id="echoid-s71" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>. or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>. or not. <lb/> +&c. +</s> +</p> +<pb file="add_6782_f035v" o="35v" n="70"/> +<pb file="add_6782_f036" o="36" n="71"/> +<div xml:id="echoid-div14" type="page_commentary" level="2" n="14"> +<p> +<s xml:id="echoid-s72" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s72" xml:space="preserve"> +Columns 1 to 10 of Pascal's triangle, continued in the lower half of the page as far as column 18. <lb/> +The table is continued to column 24 on Add MS 6782, f. 37. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head8" xml:space="preserve"> +Of combinations +</head> +<p> +<s xml:id="echoid-s74" xml:space="preserve"> +These nombers are also the <lb/> +nombers which are used for the <lb/> +extraction of rootes. +</s> +</p> +<pb file="add_6782_f036v" o="36v" n="72"/> +<pb file="add_6782_f037" o="37" n="73"/> +<div xml:id="echoid-div15" type="page_commentary" level="2" n="15"> +<p> +<s xml:id="echoid-s75" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s75" xml:space="preserve"> +Pascal's triangle as far as column 24, continued from Add MS 6782, f. 36. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head9" xml:space="preserve"> +Of combinations +</head> +<pb file="add_6782_f037v" o="37v" n="74"/> +<pb file="add_6782_f038" o="38" n="75"/> +<div xml:id="echoid-div16" type="page_commentary" level="2" n="16"> +<p> +<s xml:id="echoid-s77" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s77" xml:space="preserve"> +On this page, as on Add MS 6782, f. 36, Harriot recognizes the usefulness of the triangular numbers +both for calculating numbers of combinations and for extraction of roots. <lb/> +In the note in the bottom right hand corner of the page, Harriot mentions Boethius, Jordanus, and Maurolico +as writers on figurate numbers. +His source for both Boethius and Jordanus was probably Jacques Lefevre d'Etaples (Jacob Faber Satuplensis), +<emph style="it">Epitome, compendiosaque introductio in libros arithmeticos diui Severini Boetij</emph> (1503, 1522), +which includes a comparison of the 'De instituione' of Boethius with the 'De arithmetica' of Jordanus. +His source for Maurolico was the <emph style="it">Arithmeticorum libri duo</emph> (1575), +which he cited several times elsewhere. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head10" xml:space="preserve"> +Of combinations +</head> +<p> +<s xml:id="echoid-s79" xml:space="preserve"> +A Generall rule to get the mayne summe of all the complications <lb/> +of any Nomber of Species without the table of combinations. +</s> +</p> +<p> +<s xml:id="echoid-s80" xml:space="preserve"> +According to the nomber of species: understand as many termes <lb/> +to be gotten in continuall proportion or progression, beginning at <lb/> +a unite & making every terme double to his precedent: the +<emph style="super">double of the</emph> last <lb/> +terme lesse by a unite is the summe desired. or the somme of the <lb/> +progression. +</s> +</p> +<p> +<s xml:id="echoid-s81" xml:space="preserve"> +As for example. +</s> +<s xml:id="echoid-s82" xml:space="preserve"> +I wold know all the complications <lb/> +of 6. species. together with the nomber of the simples. <lb/> +</s> +<s xml:id="echoid-s83" xml:space="preserve"> +the sixth terme of such a progression I spake of, <lb/> +is 32. +</s> +<s xml:id="echoid-s84" xml:space="preserve"> +<emph style="st">therefore</emph> +<emph style="super">The double lesse by a unite is</emph> 63, +<emph style="st">is</emph> the summe of all the <emph style="st">the</emph> <lb/> +complications with the nomber of simples which were <lb/> +sought. +</s> +<lb/> +<s xml:id="echoid-s85" xml:space="preserve"> +If the number of species be greate; the last terme <lb/> +desired is to be gotten by <emph style="super">the</emph> rule of progression in <lb/> +arithmeticke. +</s> +<lb/> +<s xml:id="echoid-s86" xml:space="preserve"> +The reason of the rule is easily to be conceaved <lb/> +out of the particular constructions in an other <lb/> +paper annexed. +<sc> +The 'other paper' referred to here is probably Add MS 6782, f. 331. +</sc> +</s> +</p> +<p> +<s xml:id="echoid-s87" xml:space="preserve"> +A Generall methode for the particular summes of <lb/> +complications : +</s> +</p> +<p> +<s xml:id="echoid-s88" xml:space="preserve"> +As for example of 6. +</s> +<s xml:id="echoid-s89" xml:space="preserve"> +<emph style="st">I would know all</emph> first in 6 there are <lb/> +6 diverse simple species. +</s> +<s xml:id="echoid-s90" xml:space="preserve"> +Then I wold know how many <lb/> +complications of 2 wilbe found in 6; also how many of 3. & 4. &c. +</s> +</p> +<p> +<s xml:id="echoid-s91" xml:space="preserve"> +The Theoreme for the rule is this: <lb/> +</s> +</p> +<p> +<s xml:id="echoid-s92" xml:space="preserve"> +As 2 hath in proportion to the second nomber from the nomber <lb/> +of species towardes a unite: so hath the nomber of variety of unites <lb/> +to the somme of the complications of 2. +</s> +<lb/> +<s xml:id="echoid-s93" xml:space="preserve"> +And as 3 hath in proportion to the third nomber from the nomber <lb/> +of species towardes a unite: so hath the nomber of <emph style="st">variety of</emph> compli-<lb/> +cations of 2, last gotten; to the somme of the complications of 3. <lb/> +</s> +<s xml:id="echoid-s94" xml:space="preserve"> +& so forth, <emph style="st">generally</emph> as wilbe manifest by the example following. +</s> +</p> +<p> +<s xml:id="echoid-s95" xml:space="preserve"> +An example for 20. +</s> +</p> +<p> +<s xml:id="echoid-s96" xml:space="preserve"> +The practice is playne. <lb/> +</s> +<s xml:id="echoid-s97" xml:space="preserve"> +The theoreme is to be <lb/> +demonstrated out of <lb/> +Boetius or Maurolicus, <lb/> +& I thinke Jordanus. <lb/> +</s> +<s xml:id="echoid-s98" xml:space="preserve"> +by the doctrine of genera- <lb/> +ting triangular nombers <lb/> +& of triangular, piramidal. <lb/> +& of piramidal, triangle- <lb/> +pyramidal &c. <lb/> +</s> +<s xml:id="echoid-s99" xml:space="preserve"> +And is worth the noting <lb/> +for some other respects <lb/> +especially <emph style="st">of generating</emph> <lb/> +for getting the nomber <lb/> +of [¿]complicity[?] that <lb/> +belongs to any dignityes <lb/> +for extracting there roote; <lb/> +seeing those nombers are <lb/> +the very same. +</s> +</p> +<pb file="add_6782_f038v" o="38v" n="76"/> +<pb file="add_6782_f039" o="39" n="77"/> +<div xml:id="echoid-div17" type="page_commentary" level="2" n="17"> +<p> +<s xml:id="echoid-s100" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s100" xml:space="preserve"> +By 'transpositions' Harriot means what we would now call permutations. +His 'single variations' are what we would now call cyclic permutations. +For simple diagrams illustrating cyclic permutations see Add MS 6782, f. 43v and f. 225v. <lb/> +On this folio he lists all possible permutations of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>, +and begins a list for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi><mi>e</mi></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head11" xml:space="preserve"> +Of Transpositions +</head> +<p> +<s xml:id="echoid-s102" xml:space="preserve"> +Single variations. +</s> +</p> +<p> +<s xml:id="echoid-s103" xml:space="preserve"> +They are <lb/> +so many as <lb/> +there are <lb/> +species. +</s> +</p> +<p> +<s xml:id="echoid-s104" xml:space="preserve"> +The nomber of transpositions of any nomber of species being given: The nomber <lb/> +of transpositions of the next nomber of species, is a nomber that riseth of there <lb/> +multiplication. +</s> +</p> +<p> +<s xml:id="echoid-s105" xml:space="preserve"> +For: suppose the nomber of transpositions of 3 species, that is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, to be 6. The <lb/> +next nomber to be transposed is 4. which let be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>. +</s> +<s xml:id="echoid-s106" xml:space="preserve"> +Now <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, in respect of <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math> hath foure places. that is he may be next after <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>: after <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>: after <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>: or <lb/> +before <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. +</s> +<s xml:id="echoid-s107" xml:space="preserve"> +So many places it hath with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mi>b</mi></mstyle></math>, & the rest of the 6. +</s> +<s xml:id="echoid-s108" xml:space="preserve"> +Therefore 4 times <lb/> +6, which is 24. is the nomber of transpositions of 4 species. +</s> +<s xml:id="echoid-s109" xml:space="preserve"> +The like reason <lb/> +is of all others that <emph style="st">may</emph> follow +<foreign xml:lang="lat">in infinitum</foreign>. +</s> +</p> +<p> +<s xml:id="echoid-s110" xml:space="preserve"> +Transpositions of: +</s> +</p> +<p> +<s xml:id="echoid-s111" xml:space="preserve"> +Number of <lb/> +Termes of <lb/> +variations +</s> +</p> +<p> +<s xml:id="echoid-s112" xml:space="preserve"> +Sume of the <lb/> +species totall +</s> +</p> +<pb file="add_6782_f039v" o="39v" n="78"/> +<pb file="add_6782_f040" o="40" n="79"/> +<div xml:id="echoid-div18" type="page_commentary" level="2" n="18"> +<p> +<s xml:id="echoid-s113" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s113" xml:space="preserve"> +This small table shows all the combinations and permutations of up to 7 letters. +The figures in the column under 7, for example, show all the combinations of 7 single letters (7), +all the combinations of 2 letters (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mo>×</mo><mn>6</mn><mo>=</mo><mn>4</mn><mn>2</mn></mstyle></math>), +of 3 letters (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mo>×</mo><mn>6</mn><mo>×</mo><mn>5</mn><mo>=</mo><mn>2</mn><mn>1</mn><mn>0</mn></mstyle></math>), and so on. <lb/> +Lines drawn between columns show how figures in a given column are obtained +from those in the preceding column. +The figures in column 7, for example, are obtained from those in column 6 by multiplying by 7. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head12" xml:space="preserve"> +combinations & transpositions together. +</head> +<pb file="add_6782_f040v" o="40v" n="80"/> +<div xml:id="echoid-div19" type="page_commentary" level="2" n="19"> +<p> +<s xml:id="echoid-s115" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s115" xml:space="preserve"> +Tables showing the most likely sums on 5 dice, or on 6 dice (totals only). <lb/> +The left hand table is a frequency table for the sums on five dice, +constructed by adding copies of the columns of table (v) from Add MS 6782, f. 40 +(just as table (v) there was constructed from table (iv). <lb/> +The two right hand columns are a frequency table for the sums on six dice. +Here the individual columns that make up the sum have not been written down but +the additions have been carried out in the working at the bottom of the page. +In the table itself Harriot has written only the totals. <lb/> +As on Add MS 6782, f. 40, the most likely sums are marked with crosses. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f041" o="41" n="81"/> +<div xml:id="echoid-div20" type="page_commentary" level="2" n="20"> +<p> +<s xml:id="echoid-s117" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s117" xml:space="preserve"> +The two words 'on dice' in the title are written in Harriot's Algonquin alphabet +(see Add MS 6782, f. 337). <lb/> +The two tables at the top of the page show the sums that can be obtained on +(i) two dice (sums ranging from 2 to 12) or (ii) three dice (sums ranging from 3 to 18). <lb/> +The three tables in the lower part of the page are frequency tables for the sums on +(iii) two dice; (iv) three dice ; (v) four dice. <lb/> +Tables (iii) and (iv) can be calculated directly from (i) and (ii). +However, the layout shows that (iv) can also be calculated by taking copies of the totals from (iii), +staggering their starting position, and then adding; +this is equivalent to taking the totals from (iii) +and then adding 1, 2, 3, 4, 5, 6, in turn to represent the throw of the third dice. <lb/> +For each of (iii), (iv), and (v) the most likely sums are marked with a small cross. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head13" xml:space="preserve"> +Of combinations & transpositions of the numbers [on diz] +</head> +<pb file="add_6782_f041v" o="41v" n="82"/> +<pb file="add_6782_f042" o="42" n="83"/> +<div xml:id="echoid-div21" type="page_commentary" level="2" n="21"> +<p> +<s xml:id="echoid-s119" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s119" xml:space="preserve"> +Another version of the tables from the lower half of Add MS 6782, f. 81. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f042v" o="42v" n="84"/> +<div xml:id="echoid-div22" type="page_commentary" level="2" n="22"> +<p> +<s xml:id="echoid-s121" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s121" xml:space="preserve"> +The word 'diz' (dice) is written at the top of the page in Harriot's Algonquin alphabet +(see Add MS 6782, f. 337.) +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p> +<s xml:id="echoid-s123" xml:space="preserve"> +[diz] +</s> +</p> +<pb file="add_6782_f043" o="43" n="85"/> +<pb file="add_6782_f043v" o="43v" n="86"/> +<pb file="add_6782_f044" o="44" n="87"/> +<div xml:id="echoid-div23" type="page_commentary" level="2" n="23"> +<p> +<s xml:id="echoid-s124" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s124" xml:space="preserve"> +This folio quotes some text from Girolam Cardano, <emph style="it">Opus novum de proportionibus</emph> +(1570), page 187, Proposition 170. The copy is in an unknown hand. +The table below the text is exactly as given by Cardano. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head14" xml:lang="lat"> +Cardanus de proportionibus. prop. 170. +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s126" xml:space="preserve"> +Ut autem habeas numeros singulorum ordinum, in quavis multitudino, deducito <lb/> +numerum ordinis a primo, & divide per numerum ordinis ipsius reliquum, <lb/> +et illud quod proventi, ducito in numerum maximum praecedentis ordinis, <lb/> +et habebis numerum quaesitum. </s> +<s xml:id="echoid-s127" xml:space="preserve"> +Velut si sint undecim, volo scire breviter numeros, <lb/> +qui fiunt ex variatione trium. </s> +<s xml:id="echoid-s128" xml:space="preserve"> +Primum deduco pro secundo ordine 1 ex 11 fit 10, <lb/> +divido per 2 numerum ordinis, exit 5, duco in 11 fit 55 numerus secundi ordinis. Inde <lb/> +detraho 2, qui est numerus differentiae ordinis totij a primo ex 11, relinquitur 9, <lb/> +divido 9 per 3 numerum ordinis exit 3, duco 3 in 55 numerum secundi fit 165, <lb/> +numerus totij ordinis. </s> +<s xml:id="echoid-s129" xml:space="preserve"> +Similiter volo numerum variatione quatuor, <lb/> +dedco 3 differentiam 4 a primo ordine ab 11 relinquitur 8, divido 8 per 4 <lb/> +numero ordinis, exit 2, duco 2 in 195 fit 330, numeri quarti ordinis. </s><lb/> +<s xml:id="echoid-s130" xml:space="preserve"> +Similiter pro quinto detraho 4 differentiam a primo ordine, relinquitur 7, <lb/> +divido per 5 numerum ordinis exit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mfrac><mrow><mn>2</mn></mrow><mrow><mn>5</mn></mrow></mfrac></mstyle></math>, duco in 330 numero praecedentis <lb/> +ordine, fit 462 numerus quinti ordinis. +<lb/>[<emph style="it">tr: +As moreover you have the numbers of a single row, of any size, +you can derive the numbers of each place from the first, +and divide by the number of the place, and what arises multiply by the greatest number +of the preceding place,and you will have the number sought. +Thus if there are eleven [objects], +I want to know quickly the number that arises for three choices. First, for the second place, +I take 1 from 11 which makes 10, I divide by 2, the number of the place, there comes out 5, +I multiply by 11 to make 55, the number of the second place. Next I subtract 2, +which is the number of the difference of all the places from the first, from 11, there remains 9, +I divide 9 by 3, +the number of the place, there comes out 3, I multiply 3 by 55, the second number, to make 165, +the number in the third place. Similarly if I want the number for four choices, I take 3, +the difference of 4 from the first place, from 11, there is left 8, I divide 8 by 4, +the number of the place, there comes out 2, I multiply 2 by 165 to make 330, +the number of the fourth place. +Similarly for the fifth I subtract 4, the difference from the first place, there remains 7, I divide by 5, +the number of the place, there comes out <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mfrac><mrow><mn>2</mn></mrow><mrow><mn>5</mn></mrow></mfrac></mstyle></math>, I multiply by 330, +the number of the previous place, to make 462, the number of the fifth place. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f044v" o="44v" n="88"/> +<pb file="add_6782_f045" o="45" n="89"/> +<div xml:id="echoid-div24" type="page_commentary" level="2" n="24"> +<p> +<s xml:id="echoid-s131" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s131" xml:space="preserve"> +This folio appears to deal with 6 dice numbered as follows: +</s> +<lb/> +<s xml:id="echoid-s132" xml:space="preserve"> +first die: 0 0 0 0 0 1 <lb/> +second die: 0 0 0 0 0 2 <lb/> +third die: 0 0 0 0 0 3 <lb/> +fourth die: 0 0 0 0 0 4 <lb/> +fifth die: 0 0 0 0 0 5 <lb/> +sixth die: 0 0 0 0 0 6 +</s> +<lb/> +<s xml:id="echoid-s133" xml:space="preserve"> +The tables show the possible outcomes of throwing the first, then the first and the second, +then the first and the second and the third, and so on. +In the table for six dice, for example, we see that combinations with one 0 and five other numbers +can appear in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mo>×</mo><mn>6</mn><mo>=</mo><mn>3</mn><mn>0</mn></mstyle></math> ways (since any of the five zeros can appear in any of the six positions). <lb/> +Squeezed below and between the main tables are frequency tables showing how many times each sum can appear. +For six dice, for example, we see that the total 3 can arise in two ways (as 1 + 2 or as 0 + 3) +giving 3,750 possibilities in total. +In each of these tables the number of possibilites is summed, giving the appropriate power of 6 in each case. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f045v" o="45v" n="90"/> +<pb file="add_6782_f046" o="46" n="91"/> +<pb file="add_6782_f046v" o="46v" n="92"/> +<pb file="add_6782_f047" o="47" n="93"/> +<pb file="add_6782_f047v" o="47v" n="94"/> +<pb file="add_6782_f048" o="48" n="95"/> +<pb file="add_6782_f048v" o="48v" n="96"/> +<div xml:id="echoid-div25" type="page_commentary" level="2" n="25"> +<p> +<s xml:id="echoid-s135" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s135" xml:space="preserve"> +Combinations of quantities generated by multiplication. <lb/> +The letters <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> stand for +<foreign xml:lang="lat">pondus</foreign> (weight), +<foreign xml:lang="lat">magnitudo</foreign> (magnitude), +<foreign xml:lang="lat">figura</foreign> (area), +<foreign xml:lang="lat">situs</foreign> (place), +<foreign xml:lang="lat">altitudo</foreign> (altitude) +(see Add MS 6786, f. 291). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f049" o="49" n="97"/> +<div xml:id="echoid-div26" type="page_commentary" level="2" n="26"> +<p> +<s xml:id="echoid-s137" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s137" xml:space="preserve"> +The tables on this folio appear to have been begun at the top left +but then re-started and continued along the right-hand edge. <lb/> +The tables are calculated in turn for 1, 2, 3, 4, 5, 6 throws of a die. <lb/> +Take, for example, the fourth table, for four throws of a die. <lb/> +The first row indicates that the combination 1111 can occur in only one way. <lb/> +The next two rows indicate how many ways only 1 and 2 can occur, distributed as either +3 + 1 (thus, 1112, 1121, 1211, 2111, 2221, 2212, 2122, 1222), that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mo>×</mo><mn>4</mn><mo>=</mo><mn>8</mn></mstyle></math> ways in total, +or as +2 + 2 (thus, 1122, 1212, 1221, 2112, 2121, 2211), that is, 3 + 3 = 6 ways in total. +These two calculations are shown in full on Add MS 6782, f. 50v. <lb/> +The fourth row indicates how many ways only 1, 2, and 3 can occur, +with any one of them appearing twice (thus 1123, 1132, 1212, 3112, ...), that is, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>3</mn><mo>=</mo><mn>3</mn><mn>6</mn></mstyle></math> ways in total. +Further details of the calculation are shown on Add MS 6782, f. 50v. <lb/> +The fifth and final row indicates how many ways 1, 2, 3, 4 can appear, that is, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>4</mn><mo>=</mo><mn>2</mn><mn>4</mn></mstyle></math> ways in total. <lb/> +All the other tables are calculated in a similar way. +Several of the calculations can be seen on Add MS 6782, f. 50v. <lb/> +Below the line (still reading the page sideways) are two further tables; +for the continuation of these, see Add MS 6782, f. 50. <lb/> +Harriot also includes some brief notes to explain how the tables have been derived. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p> +<s xml:id="echoid-s139" xml:space="preserve"> +11112 variatur per: +<lb/>[<emph style="it">tr: +11112 may be varied by: +</emph>]<lb/> +</s> +</p> +<p> +<s xml:id="echoid-s140" xml:space="preserve"> +conversionem, ut +22221. +<lb/>[<emph style="it">tr: +conversion, as 22221. +</emph>]<lb/> +</s> +</p> +<p> +<s xml:id="echoid-s141" xml:space="preserve"> +transpositionem, <lb/> +11112, <lb/> +11121, <lb/> +11211, <lb/> +12111, <lb/> +21111 +<lb/>[<emph style="it">tr: +transposition, <lb/> +11112, <lb/> +11121, <lb/> +11211, <lb/> +12111, <lb/> +21111 +</emph>]<lb/> +</s> +</p> +<p> +<s xml:id="echoid-s142" xml:space="preserve"> +coniugationum ut <emph style="st">supra</emph> <lb/> +sunt 2 ex 6; sunt 15<emph style="super">ies</emph> +<lb/>[<emph style="it">tr: +conjugation, as there are 2 out of 6, there are 15 ways +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f049v" o="49v" n="98"/> +<pb file="add_6782_f050" o="50" n="99"/> +<div xml:id="echoid-div27" type="page_commentary" level="2" n="27"> +<p> +<s xml:id="echoid-s143" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s143" xml:space="preserve"> +On this folio Harriot uses the totals he has arrived at on Add MS 6783, f. 49, +but now extends the calculations to all six numbers on the die. <lb/> +As for Add MS 6783, f. 49, we will once again examine the fourth table. <lb/> +The first row indicates that a given number can be appear four times in just one way; +but the given number can be chosen in six ways, so there are <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>×</mo><mn>6</mn><mo>=</mo><mn>6</mn></mstyle></math> such combinations in total. <lb/> +The second row indicates that any two numbers can occur together, with 3 of one and 1 of the other, in 8 ways +(as calculated on Add MS 6783, f. 49,); +but two numbers can be chosen from six in 15 ways (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>6</mn><mo>×</mo><mn>5</mn></mrow><mrow><mn>2</mn><mo>×</mo><mn>1</mn></mrow></mfrac></mstyle></math>), +so there are <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>8</mn><mo>×</mo><mn>5</mn><mo>=</mo><mn>1</mn><mn>2</mn><mn>0</mn></mstyle></math> such combinations in total. <lb/> +The third row indicates that any two numbers can occur together, with 2 of one and 2 of the other, in 6 ways +(as calculated on Add MS 6783, f. 49,); +but two numbers can be chosen from six in 15 ways (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>6</mn><mo>×</mo><mn>5</mn></mrow><mrow><mn>2</mn><mo>×</mo><mn>1</mn></mrow></mfrac></mstyle></math>), +so there are <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mo>×</mo><mn>1</mn><mn>5</mn><mo>=</mo><mn>9</mn><mn>0</mn></mstyle></math> such combinations in total. <lb/> +The fourth row indicates that any three numbers can occur together in 36 ways +(as calculated on Add MS 6783, f. 49,); +but three numbers can be chosen from six in 20 ways (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>6</mn><mo>×</mo><mn>5</mn><mo>×</mo><mn>4</mn></mrow><mrow><mn>3</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>1</mn></mrow></mfrac></mstyle></math>), +so there are <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mn>6</mn><mo>×</mo><mn>2</mn><mn>0</mn><mo>=</mo><mn>7</mn><mn>2</mn><mn>0</mn></mstyle></math> such combinations in total. <lb/> +The fifth row indicates that any four numbers can occur together in 24 ways +(as calculated on Add MS 6783, f. 49,); +but four numbers can be chosen from six in 15 ways (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>6</mn><mo>×</mo><mn>5</mn><mo>×</mo><mn>4</mn><mo>×</mo><mn>3</mn></mrow><mrow><mn>4</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>1</mn></mrow></mfrac></mstyle></math>), +so there are <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>4</mn><mo>×</mo><mn>1</mn><mn>5</mn><mo>=</mo><mn>3</mn><mn>6</mn><mn>0</mn></mstyle></math> such combinations in total. <lb/> +As a check on the calculations, Harriot has calculated powers of 6 on the same page. +It is easily seen from this that each table includes all the possibilities for that number of throws. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f050v" o="50v" n="100"/> +<div xml:id="echoid-div28" type="page_commentary" level="2" n="28"> +<p> +<s xml:id="echoid-s145" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s145" xml:space="preserve"> +Calculations for Add MS 6782, f. 97. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f051" o="51" n="101"/> +<pb file="add_6782_f051v" o="51v" n="102"/> +<pb file="add_6782_f052" o="52" n="103"/> +<pb file="add_6782_f052v" o="52v" n="104"/> +<pb file="add_6782_f053" o="53" n="105"/> +<pb file="add_6782_f053v" o="53v" n="106"/> +<pb file="add_6782_f054" o="54" n="107"/> +<pb file="add_6782_f054v" o="54v" n="108"/> +<pb file="add_6782_f055" o="55" n="109"/> +<div xml:id="echoid-div29" type="page_commentary" level="2" n="29"> +<p> +<s xml:id="echoid-s147" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s147" xml:space="preserve"> +For the definition of binomes of the third kind, see Add MS 6782, f. 267. <lb/> +On this page, Harriot shows that the cube of a binome of the third kind is again a binome of the third kind. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head15" xml:space="preserve" xml:lang="lat"> +De cubo binomij 3<emph style="super">i</emph> +<lb/>[<emph style="it">tr: +On the cube of a third binome +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s149" xml:space="preserve"> +<reg norm="binomij" type="abbr">bin</reg>. 3. +<lb/>[<emph style="it">tr: +a binome of the third kind. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s150" xml:space="preserve"> +<reg norm="binomij" type="abbr">bin</reg>. 1. +<lb/>[<emph style="it">tr: +a binome of the first kind. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s151" xml:space="preserve"> +Ergo cubus: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>5</mn><mn>4</mn><mn>0</mn><mn>8</mn></mrow></msqrt><mo>+</mo><msqrt><mrow><mn>5</mn><mn>4</mn><mn>0</mn><mn>0</mn></mrow></msqrt></mstyle></math>. <reg norm="binomij" type="abbr">bin</reg>. 3. +<reg norm="differentia" type="abbr">diff</reg>. +<reg norm="quadrati" type="abbr">quad</reg>: <lb/> +8. cubus. +<lb/>[<emph style="it">tr: +Therefore the cube <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>5</mn><mn>4</mn><mn>0</mn><mn>8</mn></mrow></msqrt><mo>+</mo><msqrt><mrow><mn>5</mn><mn>4</mn><mn>0</mn><mn>0</mn></mrow></msqrt></mstyle></math> is a binome of the third kind; +the difference of the squares <lb/> +is 8, a cube.</emph>]<lb/> +</s> +<lb/> +</p> +<pb file="add_6782_f055v" o="55v" n="110"/> +<pb file="add_6782_f056" o="56" n="111"/> +<div xml:id="echoid-div30" type="page_commentary" level="2" n="30"> +<p> +<s xml:id="echoid-s152" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s152" xml:space="preserve"> +For the definition of binomes of the fourth kind, see Add MS 6782, f. 267. <lb/> +On this page, Harriot shows that the cube of a binome of the fourth kind is again a binome of the fourth kind. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head16" xml:space="preserve" xml:lang="lat"> +De cubo binomij 4<emph style="super">i</emph> +<lb/>[<emph style="it">tr: +On the cube of a fourth binome +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s154" xml:space="preserve"> +<reg norm="binomij" type="abbr">bin</reg>. 4. +<lb/>[<emph style="it">tr: +a binome of the fourth kind. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s155" xml:space="preserve"> +<reg norm="binomij" type="abbr">bin</reg>. 1. +<lb/>[<emph style="it">tr: +a binome of the first kind. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s156" xml:space="preserve"> +Ergo cubus <reg norm="binomij" type="abbr">bin</reg>. 4. +<lb/>[<emph style="it">tr: +Therefore the cube is a binome of the fourth kind; +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s157" xml:space="preserve"> +512. <reg norm="differentia" type="abbr">dra</reg> <reg norm="quadrati" type="abbr">quad</reg>: <lb/> +cub +<lb/>[<emph style="it">tr: +512. the difference of the squares, is a cube.</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s158" xml:space="preserve"> +Aliter +<lb/>[<emph style="it">tr: +Another way +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s159" xml:space="preserve"> +Ergo cubus <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mn>2</mn><mn>0</mn><mo>+</mo><msqrt><mrow><mn>5</mn><mn>1</mn><mn>7</mn><mo>,</mo><mn>8</mn><mn>8</mn><mn>8</mn></mrow></msqrt></mstyle></math> <lb/> +Ut supra. +<lb/>[<emph style="it">tr: +Therefore the cube is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mn>2</mn><mn>0</mn><mo>+</mo><msqrt><mrow><mn>5</mn><mn>1</mn><mn>7</mn><mo>,</mo><mn>8</mn><mn>8</mn><mn>8</mn></mrow></msqrt></mstyle></math>. As above. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f056v" o="56v" n="112"/> +<pb file="add_6782_f057" o="57" n="113"/> +<div xml:id="echoid-div31" type="page_commentary" level="2" n="31"> +<p> +<s xml:id="echoid-s160" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s160" xml:space="preserve"> +The table at the top left shows the number of pathways from centre to corner +for squares of size <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>×</mo><mn>1</mn></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>1</mn><mo>×</mo><mn>2</mn><mn>1</mn></mstyle></math>. For a <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mo>×</mo><mn>5</mn></mstyle></math> square, for instance, +there are 20 pathways to each corner, and so 80 in all. +(There is an error in the calculation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>9</mn><mo>×</mo><mn>9</mn></mstyle></math> squares, where there are 70 pathways to each corner, +making a total of 280 in all, not 240.) <lb/> +Below the table are the numbers of pathways for the squares on Add MS 6782, f. 28 and f. 27, +SILO PRINCEPS FECIT (17 letters, 51,480 pathways) and HENRICUS PRINCEPS FECIT (21 letters, 739,024 pathways). <lb/> +The calculations down the right hand side of the page show the multiplication +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>1</mn><mo>×</mo><mn>1</mn><mn>2</mn><mo>×</mo><mn>1</mn><mn>3</mn><mo>×</mo><mo>…</mo><mo>×</mo><mn>2</mn><mn>1</mn></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s162" xml:space="preserve"> +Silo princeps fecit (17) 51,480 +<lb/>[<emph style="it">tr: +Prince Silo made it +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s163" xml:space="preserve"> +Jacobus +</s> +<lb/> +<s xml:id="echoid-s164" xml:space="preserve"> +Henricus princeps fecit (21) 739,024 +<lb/>[<emph style="it">tr: +Prince Henry made it +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s165" xml:space="preserve"> +Carolus princeps fecit (20) +<lb/>[<emph style="it">tr: +Prince Charles made it +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f057v" o="57v" n="114"/> +<div xml:id="echoid-div32" type="page_commentary" level="2" n="32"> +<p> +<s xml:id="echoid-s166" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s166" xml:space="preserve"> +The upper half of the page shows calculations of the number of pathways through a quarter square +for up to 9 letters. <lb/> +The calculation in the lower half of the page arrives at the total 705,432. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f058" o="58" n="115"/> +<div xml:id="echoid-div33" type="page_commentary" level="2" n="33"> +<p> +<s xml:id="echoid-s168" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s168" xml:space="preserve"> +This folios shows a version of Pascal's triangle, with the numbers 1, 2, 6, 20, 70, ... +along the diagonal emphasized. <lb/> +Here the numbers represent the number of pathways through a square with up to 144 cells, +from the starting point (marked 0) to any cell in the grid. +The numbers along the diagonal show the number of pathways from corner to corner, +as required on Add MS 6782, f. 27 and f. 28. +The numbers 12,870 and 184,756 from f. 28 and f. 27 appear on this diagonal, +as does the number 705,432 calculated on Add MS 6782, f. 57v. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s170" xml:space="preserve"> +<reg norm="examinatur" type="abbr">examinat</reg>. +<lb/>[<emph style="it">tr: +examined +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f058v" o="58v" n="116"/> +<pb file="add_6782_f059" o="59" n="117"/> +<pb file="add_6782_f059v" o="59v" n="118"/> +<pb file="add_6782_f060" o="60" n="119"/> +<div xml:id="echoid-div34" type="page_commentary" level="2" n="34"> +<p> +<s xml:id="echoid-s171" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s171" xml:space="preserve"> +Quarter squares, completed with numbers. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f060v" o="60v" n="120"/> +<pb file="add_6782_f061" o="61" n="121"/> +<pb file="add_6782_f061v" o="61v" n="122"/> +<pb file="add_6782_f062" o="62" n="123"/> +<pb file="add_6782_f062v" o="62v" n="124"/> +<pb file="add_6782_f063" o="63" n="125"/> +<div xml:id="echoid-div35" type="page_commentary" level="2" n="35"> +<p> +<s xml:id="echoid-s173" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s173" xml:space="preserve"> +The table on this folio shows the same information as Add MS 6782, f. 58, +but now in the form of one quarter of a word square. +The 'letters' in each cell are the upper entries, in slightly heavier ink. +The lower entry in each cell is the number of pathways to that square, starting from the top left hand corner. <lb/> +Grid squares along the diagonal have been slightly shaded for emphasis. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f063v" o="63v" n="126"/> +<pb file="add_6782_f064" o="64" n="127"/> +<div xml:id="echoid-div36" type="page_commentary" level="2" n="36"> +<p> +<s xml:id="echoid-s175" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s175" xml:space="preserve"> +Squares of size <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>×</mo><mn>1</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mo>×</mo><mn>3</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mo>×</mo><mn>5</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mo>×</mo><mn>7</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>9</mn><mo>×</mo></mstyle></math>, +completed with numbers. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f064v" o="64v" n="128"/> +<pb file="add_6782_f065" o="65" n="129"/> +<pb file="add_6782_f065v" o="65v" n="130"/> +<pb file="add_6782_f066" o="66" n="131"/> +<pb file="add_6782_f066v" o="66v" n="132"/> +<pb file="add_6782_f067" o="67" n="133"/> +<div xml:id="echoid-div37" type="page_commentary" level="2" n="37"> +<p> +<s xml:id="echoid-s177" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s177" xml:space="preserve"> +In this page. Harriot begins by constructing a table of the interest paid after seven years +on a capital sum of £<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math> at an annual rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>. <lb/> +The first row shows the total if the interest is paid yearly (7 payments). <lb/> +The second row shows the total if interest is paid twice a year (14 payments). <lb/> +The third row shows the total if interest is paid three times a year (21 payments). <lb/> +From here, Harriot immediately generalizes to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> payments per year. +He then (implicitly) allows <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> to become very large, indeed infinitely large, +so that the fractions <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mo maxsize="1">(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo maxsize="1">)</mo><mi>n</mi></mrow><mrow><mi>n</mi><mi>n</mi></mrow></mfrac></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mo maxsize="1">(</mo><mi>n</mi><mo>-</mo><mn>2</mn><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo maxsize="1">)</mo><mi>n</mi></mrow><mrow><mi>n</mi><mi>n</mi><mi>n</mi></mrow></mfrac></mstyle></math>, ... can all be taken to be 1. +Harriot then substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mn>1</mn><mn>0</mn></mstyle></math> to obtain the interest on £100 at a rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mstyle></math>, +paid continuously over seven years. The total comes to £201 7 shillings and 6 pence, +plus a further fraction that he estimates is not quite <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>7</mn></mrow><mrow><mn>1</mn><mn>0</mn><mn>0</mn></mrow></mfrac></mstyle></math> pence. +(There were 20 shillings (s) to £1, and 12 pence (d) to 1 shillling.) <lb/> +This page combines the calculations on the nearby f. 68 +(interest on £<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> at an annual rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>, paid at decreasing intervals) +with those f. 69 +(interest on £100 at an annual rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mstyle></math> paid at decreasing intervals and taken to a limit. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head17" xml:space="preserve"> +Interest upon interest for 7 yeares. +</head> +<p> +<s xml:id="echoid-s179" xml:space="preserve"> +The sum of interest upon interest <lb/> +continually for every instant <emph style="super">in</emph> seven <lb/> +yeares with the principall of 100£ <lb/> +after the rate of 10 in the 100 for <lb/> +a yeare. +</s> +</p> +<p> +<s xml:id="echoid-s180" xml:space="preserve"> +201£ + 7s + 6d + <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>0</mn><mn>6</mn><mn>2</mn><mn>0</mn><mn>5</mn><mi>a</mi></mrow><mrow><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn></mrow></mfrac></mstyle></math> <lb/> +not <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>7</mn></mrow><mrow><mn>1</mn><mn>0</mn><mn>0</mn></mrow></mfrac></mstyle></math> +</s> +</p> +<pb file="add_6782_f067v" o="67v" n="134"/> +<pb file="add_6782_f068" o="68" n="135"/> +<div xml:id="echoid-div38" type="page_commentary" level="2" n="38"> +<p> +<s xml:id="echoid-s181" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s181" xml:space="preserve"> +Below the rough work crossed out at the top are calculations of interest on £<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math> +at an annual rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>. <lb/> +The first table shows the interest paid every year for seven years. +The words <foreign xml:lang="lat">continue proportionales</foreign> (continued proportionals) +indicate that each row is obtained by multiplication from the previous row. +The multiplier from each row to the next is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>b</mi></mrow></mfrac><mo maxsize="1">)</mo></mstyle></math>. <lb/> +The second table shows a similar calculation but now interest is added twice yearly +and the multiplier is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>b</mi></mrow></mfrac><mo maxsize="1">)</mo></mstyle></math>. +The table shows only the first four entries and then the total after 7 years. <lb/> +The third table repeats the calculation but now interest is added three times yearly +and the multiplier is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn><mi>b</mi></mrow></mfrac><mo maxsize="1">)</mo></mstyle></math>. +The table shows only the first three entries and then the total after 7 years. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p> +<s xml:id="echoid-s183" xml:space="preserve"> +yeres +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s184" xml:space="preserve"> +continue proportionales +<lb/>[<emph style="it">tr: +continued proportionals +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f068v" o="68v" n="136"/> +<pb file="add_6782_f069" o="69" n="137"/> +<div xml:id="echoid-div39" type="page_commentary" level="2" n="39"> +<p> +<s xml:id="echoid-s185" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s185" xml:space="preserve"> +After the rough work crossed out at the top, +the table show the calculation of interest on £100 at an annual rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mstyle></math>, +paid every year for five years. <lb/> +The next section is crossed out but the reader is referred by asterisk to an expression lower down the page. +Here Harriot has written a general formula for the interest paid on £100 +at an annual rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mstyle></math> after <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> years. <lb/> +In the next calculation Harriot has assumed that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> is infinitely large, +so that the fractions <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mo maxsize="1">(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo maxsize="1">)</mo><mi>n</mi></mrow><mrow><mi>n</mi><mi>n</mi></mrow></mfrac></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mo maxsize="1">(</mo><mi>n</mi><mo>-</mo><mn>2</mn><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo maxsize="1">)</mo><mi>n</mi></mrow><mrow><mi>n</mi><mi>n</mi><mi>n</mi></mrow></mfrac></mstyle></math>, ... can all be taken to be 1. +Thus he obtains the interest on £100 at an annual rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mstyle></math> paid continuously. <lb/> +In calculating the sum, Harriot has drawn a vertical line that cuts off the calculation after 6 terms. +He notes underneath that £<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn><mn>0</mn></mrow></mfrac><mo>=</mo><mn>4</mn></mstyle></math>d (4 pence), that £<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mn>4</mn><mn>0</mn><mn>0</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mstyle></math>d, +and that the sum of all the remaining terms will not make up as much as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>2</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mstyle></math>d. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p> +<s xml:id="echoid-s187" xml:space="preserve"> +yeares +</s> +</p> +<p> +<s xml:id="echoid-s188" xml:space="preserve"> +The summe of interest upon interest <lb/> +continually for every instant <lb/> +the whole yeare with the principall <lb/> +of 100£ after the rate of 10 in ye 100 <lb/> +for the yeare. +</s> +</p> +<p> +<s xml:id="echoid-s189" xml:space="preserve"> +not <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>2</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mstyle></math> d +</s> +</p> +<pb file="add_6782_f069v" o="69v" n="138"/> +<pb file="add_6782_f070" o="70" n="139"/> +<div xml:id="echoid-div40" type="page_commentary" level="2" n="40"> +<p> +<s xml:id="echoid-s190" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s190" xml:space="preserve"> +The tables in the first row show the calculation of interest on £<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> +at an annual rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mstyle></math> for four years. <lb/> +The tables in the second row show the calculation of interest on £<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> +at an annual rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mn>0</mn></mrow></mfrac></mstyle></math> for four years. <lb/> +The tables in the third row show the calculation of interest on £<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> +at an annual rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn><mn>0</mn></mrow></mfrac></mstyle></math> for four years. <lb/> +The tables in the last row show the calculation of interest on £<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> +at an annual rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mi>c</mi></mrow></mfrac></mstyle></math> for seven years. <lb/> +The words <foreign xml:lang="lat">continue proportionales</foreign> (continued proportionals) +next to the final table indicate that each row is obtained by multiplication from the previous row. +The multiplier from each row to the next is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>c</mi></mrow></mfrac><mo maxsize="1">)</mo></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f070v" o="70v" n="140"/> +<pb file="add_6782_f071" o="71" n="141"/> +<pb file="add_6782_f071v" o="71v" n="142"/> +<pb file="add_6782_f072" o="72" n="143"/> +<pb file="add_6782_f072v" o="72v" n="144"/> +<div xml:id="echoid-div41" type="page_commentary" level="2" n="41"> +<p> +<s xml:id="echoid-s192" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s192" xml:space="preserve"> +A list of texts and page numbers. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p> +<s xml:id="echoid-s194" xml:space="preserve"> +Compendium Avicennæ. 17. <lb/> +Brevi Breviarum pro Baconis <lb/> +ad <emph style="super">[???]</emph> [???]. 95. <lb/> +Verbum abreviatum [???] <lb/> +Raymondi de Leone Vividi. 26A. <lb/> +Secretum secretorum naturæ de lazuli <lb/> +lapidis philosphonum. 285 <lb/> +Tractatus trium Bretonum Br. Baconis. <lb/> +Epistola prima. 293. <lb/> +Secunda 301. <lb/> +Tertia 314. <lb/> +Specatum secretorum. 387. +</s> +</p> +<pb file="add_6782_f073" o="73" n="145"/> +<div xml:id="echoid-div42" type="page_commentary" level="2" n="42"> +<p> +<s xml:id="echoid-s195" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s195" xml:space="preserve"> +The choice of numbers on this page suggests that it might have been written in the yeasr 1599? +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p> +<s xml:id="echoid-s197" xml:space="preserve"> +The denary Arithmetick which is in common use <lb/> +doth expresse nombers of figures in a continuall progression <lb/> +of which a unit is the first; the second is ten & may <lb/> +be termed as in a algebra a roote; the third is a hundred & <lb/> +may be termed a square. &c. +</s> +</p> +<pb file="add_6782_f073v" o="73v" n="146"/> +<pb file="add_6782_f074" o="74" n="147"/> +<div xml:id="echoid-div43" type="page_commentary" level="2" n="43"> +<p> +<s xml:id="echoid-s198" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s198" xml:space="preserve"> +At the beginning of this set of sheets Harriot has written: 'Waste papers of figurate nombers'. +They are waste only in the sense that they contain rough working. At the same time, +they show Harriot attempting something highly original, namely, +finding formulae for sequences of figurate numbers. +In modern terms, we would say he is fitting third-, fourth- or fifth-degree polynomials +to numerical sequences. <lb/> +At the top is the sequence 1, 5, 14, 30, 55, ... of sums of squares +(or of square-pyramidal numbers, see Add MS 6782, f. 155). +Just below that, Harriot has written the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>9</mn><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>1</mn><mi>a</mi><mo>+</mo><mn>2</mn><mi>a</mi></mstyle></math>, which, it seems, +is his first attempt to find a formula for the numbers in the sequence multiplied by 24 +(that is, 24, 120, 336, ...). Putting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>1</mn></mstyle></math> gives <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mo>+</mo><mn>9</mn><mo>+</mo><mn>1</mn><mn>1</mn><mo>+</mo><mn>2</mn></mstyle></math>, as required. +Putting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn></mstyle></math>, however, gives <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mo>×</mo><mn>1</mn><mn>6</mn><mo>+</mo><mn>9</mn><mo>×</mo><mn>8</mn><mo>+</mo><mn>1</mn><mn>1</mn><mo>+</mo><mn>4</mn><mo>+</mo><mn>2</mn><mo>×</mo><mn>4</mn><mo>=</mo><mn>1</mn><mn>5</mn><mn>2</mn></mstyle></math>, which is too large. +This calculation can be seen displayed vertically just below the formula. +Harriot notes that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mo>×</mo><mn>2</mn><mn>4</mn><mo>=</mo><mn>1</mn><mn>4</mn><mn>4</mn></mstyle></math> falls short of this total by 8. <lb/> +Similar trial and error calculations appear on this and several pages that follow. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head18" xml:space="preserve"> +Waste papers. <lb/> +of figurate <lb/> +nombers. +</head> +<pb file="add_6782_f074v" o="74v" n="148"/> +<div xml:id="echoid-div44" type="page_commentary" level="2" n="44"> +<p> +<s xml:id="echoid-s200" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s200" xml:space="preserve"> +Trials similar to those on f. 74, but now for the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mi>a</mi><mo>+</mo><mn>1</mn><mn>1</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>6</mn><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, +represented by the coefficients 6, 11, 6, 1. This is evaluated for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn></mstyle></math>, +giving <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mn>6</mn><mn>0</mn><mo>=</mo><mn>2</mn><mn>4</mn><mo>×</mo><mn>1</mn><mn>5</mn></mstyle></math>; +for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>5</mn></mstyle></math>, giving <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>6</mn><mn>8</mn><mn>0</mn><mo>=</mo><mn>2</mn><mn>4</mn><mo>×</mo><mn>7</mn><mn>0</mn></mstyle></math>; and (lower right) for for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn></mstyle></math>, +giving <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>2</mn><mn>0</mn><mo>=</mo><mn>2</mn><mn>4</mn><mo>×</mo><mn>5</mn></mstyle></math>. +It seems Harriot is still trying to fit the sequence 1, 5, 14, 30, ..., +which appears again alongside the coefficients 6, 11, 6, 1 on f. 76v. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f075" o="75" n="149"/> +<pb file="add_6782_f075v" o="75v" n="150"/> +<pb file="add_6782_f076" o="76" n="151"/> +<pb file="add_6782_f076v" o="76v" n="152"/> +<div xml:id="echoid-div45" type="page_commentary" level="2" n="45"> +<p> +<s xml:id="echoid-s202" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s202" xml:space="preserve"> +Further calculations with the coefficients 6, 11, 6, 1 (see f. 74v), now for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>4</mn></mstyle></math>, +giving <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>8</mn><mn>4</mn><mn>0</mn><mo>=</mo><mn>2</mn><mn>4</mn><mo>×</mo><mn>3</mn><mn>5</mn></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f077" o="77" n="153"/> +<div xml:id="echoid-div46" type="page_commentary" level="2" n="46"> +<p> +<s xml:id="echoid-s204" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s204" xml:space="preserve"> +On this folio Harriot appears to be searchng for a formula for the sequence 1, 6, 20, 50, ... +of sums of square-pyramidal numbers (see Add MS 6782, f. 155), or rather, +for those numbers multiplied 24, that is, 24, 144, 480, ... . <lb/> +Examples on the left hand side of the page test the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>8</mn><mi>a</mi><mo>+</mo><mn>4</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>0</mn><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>2</mn><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> +(note that 8 + 4 + 10 + 2 = 24). This is evaluated for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn></mstyle></math>, giving 144 (as required) +and for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn></mstyle></math>, giving 492 (too large). <lb/> +Examples in the bottom left hand corner rearrange the same coefficients in different orders: +(2, 10, 8, 4), (8, 10, 2, 4), (8, 2, 10, 4), etc. <lb/> +In examples further to the right, the fifth-degree polynomial <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>4</mn><mi>a</mi><mo>+</mo><mn>5</mn><mn>0</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>3</mn><mn>5</mn><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>0</mn><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, +represented by the coefficients 24, 50, 35, 10, 1, is evaluated for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn></mstyle></math>, giving 640, +and for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn></mstyle></math> giving 2520. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f077v" o="77v" n="154"/> +<div xml:id="echoid-div47" type="page_commentary" level="2" n="47"> +<p> +<s xml:id="echoid-s206" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s206" xml:space="preserve"> +This page shows further attempts, as on f. 77, to find coefficients that deliver +24 (when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>1</mn></mstyle></math>), 144 (when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn></mstyle></math>), and 480 (when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn></mstyle></math>). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f078" o="78" n="155"/> +<pb file="add_6782_f078v" o="78v" n="156"/> +<pb file="add_6782_f079" o="79" n="157"/> +<pb file="add_6782_f079v" o="79v" n="158"/> +<pb file="add_6782_f080" o="80" n="159"/> +<div xml:id="echoid-div48" type="page_commentary" level="2" n="48"> +<p> +<s xml:id="echoid-s208" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s208" xml:space="preserve"> +On this folio Harriot is seraching for a formula for the sequence 1, 6, 15, 28, ... of hexagonal numbers +(see Add MS 6782, f. 157), or rather, for those numbers multiplied by 24, that is, 24, 144, 360, .... <lb/> +Note that some of the coefficients on this page are negative. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f080v" o="80v" n="160"/> +<pb file="add_6782_f081" o="81" n="161"/> +<div xml:id="echoid-div49" type="page_commentary" level="2" n="49"> +<p> +<s xml:id="echoid-s210" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s210" xml:space="preserve"> +On this folio Harriot is searching for a formula for the sequence 1, 5, 12, 22, ... of pentagonal numbers +(see f. Add MS 6782, 156), or rather, for those numbers multiplied by 24, that is, 24, 120, 288, .... +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f081v" o="81v" n="162"/> +<div xml:id="echoid-div50" type="page_commentary" level="2" n="50"> +<p> +<s xml:id="echoid-s212" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s212" xml:space="preserve"> +This folio shows many examples of fifth-degree polynomials, as represented by their coefficients, +evaluated for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>5</mn></mstyle></math>. At top left, for example, +the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mn>0</mn><mi>a</mi><mo>+</mo><mn>2</mn><mn>3</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>5</mn><mn>2</mn><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>3</mn><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> is evaluated for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn></mstyle></math>, giving 840. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f082" o="82" n="163"/> +<div xml:id="echoid-div51" type="page_commentary" level="2" n="51"> +<p> +<s xml:id="echoid-s214" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s214" xml:space="preserve"> +Like f. 81v, this folio shows examples of fifth-degree polynomials, as represented by their coefficients, +evaluated for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>5</mn></mstyle></math>. At top left, for example, +the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>8</mn><mi>a</mi><mo>+</mo><mn>4</mn><mn>5</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>4</mn><mn>0</mn><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>5</mn><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>2</mn><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> is evaluated for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn></mstyle></math>, giving 3240. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f082v" o="82v" n="164"/> +<pb file="add_6782_f083" o="83" n="165"/> +<div xml:id="echoid-div52" type="page_commentary" level="2" n="52"> +<p> +<s xml:id="echoid-s216" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s216" xml:space="preserve"> +On this folio Harriot appears to be searching a formula for the sequence 1, 7, 27, 77, ... +of sums of sums of square-pyramidal numbers (see Add MS 6782, f. 155), or rather, +for those numbers multiplied by 120, that is, 120, 840, 3240, .... +For ths he needs polynmials of the fifth degree. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f083v" o="83v" n="166"/> +<pb file="add_6782_f084" o="84" n="167"/> +<div xml:id="echoid-div53" type="page_commentary" level="2" n="53"> +<p> +<s xml:id="echoid-s218" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s218" xml:space="preserve"> +The first reference in the heading is to Michael Stifel, <emph style="it">Arithmetica integra</emph> (1544), +page 15. +For Stifel, a diagonal number was obtained by multiplying the first two entries of a Pythagorean triple. +The diagonal number corresponding to the triple (3, 4, 5), for example, is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mo>×</mo><mn>4</mn><mo>=</mo><mn>1</mn><mn>2</mn></mstyle></math>. +Stifel also defined Pythagorean triples by the ratio of the two shorter sides, in this case <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math>. +He was able to write out two lists, or orders, of triples, +one with the shorter side odd (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn><mn>2</mn></mrow><mrow><mn>5</mn></mrow></mfrac></mstyle></math>, and so on), +the other with the shorter side even (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn><mn>5</mn></mrow><mrow><mn>8</mn></mrow></mfrac></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>3</mn><mn>5</mn></mrow><mrow><mn>1</mn><mn>2</mn></mrow></mfrac></mstyle></math>, and so on). +Stifel claimed that all possible triples were included in these two orders. <lb/> +The second reference in the heading, possibly added a little later, is to Johannes Praetorius (Johann Richter), +<emph style="it">Problema, quod iubet ex quatuor rectis lineis datis quadrilaterum fieri, +quod sit in circulo</emph> (1598). On the final page, +Praetorius discusses the problem of constructing cyclic quadrilaterals with rational sides. <lb/> +<lb/> +Harriot sets out to disprove Stifel's claim, by demonstrating the existence of new orders of triples. <lb/> +His first order (ordo. 1.) is the same as Stifel's first order. +The triples are set out in three columns with differences calculated between rows. +This allows Harriot to extrapolate forwards, but also backwards to a starting triple (1, 0, 1). <lb/> +The second order (ordo. 2.) is the same as Stifel's second order. +Again the triples are set out in three columns with differences calculated between rows. +As for the first order this allows Harriot to extrapolate backwards to a starting triple (4, 3, 5). +This is the first triple of the first order with the first two entries interchanged. +Perhaps this gave Harriot the idea of interchanging other pairs. +Thus he begins a third and new order (ordo. 3. novus) with (12, 5, 8), +which is the second triple from the first order with the first two entries interchanged. +This order immediately contains (20, 21, 29), which was not included in either of Stifel's orders. +The fourth order begins with (15, 8, 17), +which is the first triple from the second order with the first two entries interchanged. +And so on. +<lb/> +By the end of the page, Harriot has six orders, with differences in the left column of 2, 4, 8, 6, 10, 12, +respectively. +This seems to suggest to him a more systematic method of displaying the orders, +which he goes on to do on the next page. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head19" xml:space="preserve" xml:lang="lat"> +Examinatur Stifelius <lb/> +de numeris diagonalibus. pa. 15 <lb/> +et prætorius. pag. ult +<lb/>[<emph style="it">tr: +An examination of Stifel on diagonal numbers, page 15, and Praetorius, last page. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s220" xml:space="preserve"> +ordo. 1. <lb/> +pythag. +<lb/>[<emph style="it">tr: +Order 1, Pythagorean +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s221" xml:space="preserve"> +ord. 2. <lb/> +<reg norm="Platonic" type="abbr">platon</reg>. +<lb/>[<emph style="it">tr: +Order 2, Platonic. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s222" xml:space="preserve"> +hoc est. +<lb/>[<emph style="it">tr: +that is: +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s223" xml:space="preserve"> +Dixit quod rationes omni <lb/> +laterum sunt in istis <lb/> +duobus ordinibus. +<lb/>[<emph style="it">tr: +He said that all the ratios of sides are in these two orders. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s224" xml:space="preserve"> +Ego dico quod non. +<lb/>[<emph style="it">tr: +I say that is not so. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s225" xml:space="preserve"> +ordines sunt alij <lb/> +infiniti. +</s> +<s xml:id="echoid-s226" xml:space="preserve"> +Ut per <lb/> +sequentia patet. +<lb/>[<emph style="it">tr: +There are infinitely many other orders. As is clear from what follows. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s227" xml:space="preserve"> +ordo. 1. <lb/> +ordo. 2. <lb/> +ordo. 3. novus. +<lb/>[<emph style="it">tr: +Order 1. <lb/> +Order 2. <lb/> +Order 3, new. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s228" xml:space="preserve"> +Melior est dispositio ordinum <lb/> +in alijs chartis sequentibus. +<lb/>[<emph style="it">tr: +The arrangement of orders is better in the other sheets following. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f084v" o="84v" n="168"/> +<div xml:id="echoid-div54" type="page_commentary" level="2" n="54"> +<p> +<s xml:id="echoid-s229" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s229" xml:space="preserve"> +This folio gives a list of the hypotenuses that have been discovered on the previous page, +with the differences between them. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head20" xml:space="preserve" xml:lang="lat"> +Hypotenusorum progressio +<lb/>[<emph style="it">tr: +A progression of the hypotenuses +</emph>]<lb/> +</head> +<pb file="add_6782_f085" o="85" n="169"/> +<div xml:id="echoid-div55" type="page_commentary" level="2" n="55"> +<p> +<s xml:id="echoid-s231" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s231" xml:space="preserve"> +On this folio the orders discovered on the previous sheet (f. 84) are listed systematically. +Each new order begins with the second triple from the previous order, +with the first two entries interchanged. <lb/> +At the bottom of the page, Harriot notes the starting differences for each order. <lb/> +He has also written the enigmatic note +</s> +<lb/> +<quote xml:lang="lat"> +Hic sunt omnes primi sed hic omnes non sunt primi +</quote> +<lb/> +<s xml:id="echoid-s232" xml:space="preserve"> +which was greatly to confuse his friend Nathaniel Torporley when he came across it some years later. <lb/> +For discussion of this and the surrounding sheets see Tanner 1977. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s234" xml:space="preserve"> +1.) +</s> +</p> +<p> +<s xml:id="echoid-s235" xml:space="preserve"> +1) ordo. +<lb/>[<emph style="it">tr: +order 1 +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s236" xml:space="preserve"> +Et sic in cæteribus in infinitum. +<lb/>[<emph style="it">tr: +And so on for the rest indefinitely. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s237" xml:space="preserve"> +Hic sunt omnes primi <lb/> +sed hic omnes non sunt primi. +<lb/>[<emph style="it">tr: +Here are all the primes but here not all are prime. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s238" xml:space="preserve"> +Nota <lb/> +prima Differentia ordinis <lb/> +primi. 2. 4. sub dupla. <lb/> +Secundi. 4. 12. tripla <lb/> +Tertij. 6. 12. Dupla <lb/> +Quarti. 8. 24. tripla <lb/> +Quinti. 10. 20. Dupla <lb/> +Sexti. 12. 36. tripla <lb/> +Septimi. 14. 28. Dupla <lb/> +Octavi. 16. 48. tripla <lb/> +&c in infinitum. +<lb/>[<emph style="it">tr: +Note<lb/> +First differences of the order <lb/> +First double <lb/> +Second triple <lb/> +Third double <lb/> +Fourth triple <lb/> +Fifth double <lb/> +Sixth triple <lb/> +Seventh double <lb/> +Eighth triple <lb/> +etc. indefinitely +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f085v" o="85v" n="170"/> +<pb file="add_6782_f086" o="86" n="171"/> +<div xml:id="echoid-div56" type="page_commentary" level="2" n="56"> +<p> +<s xml:id="echoid-s239" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s239" xml:space="preserve"> +Harriot continues his orders 1 to 4 of Pythagorean triples from f. 85. +In each case the orders are continued as far as the hypotenuse closest to 1105. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s241" xml:space="preserve"> +2.) continuationes +<lb/>[<emph style="it">tr: +continuations +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s242" xml:space="preserve"> +1) ordinis +<lb/>[<emph style="it">tr: +order 1 +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s243" xml:space="preserve"> +recte +<lb/>[<emph style="it">tr: +correct +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f086v" o="86v" n="172"/> +<pb file="add_6782_f087" o="87" n="173"/> +<div xml:id="echoid-div57" type="page_commentary" level="2" n="57"> +<p> +<s xml:id="echoid-s244" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s244" xml:space="preserve"> +Harriot continues his orders 5 to 7 of Pythagorean triples from f. 85, and adds order 8. +In each case the orders are continued until the hypotenuse is equal to or greater 1105. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s246" xml:space="preserve"> +3.) +</s> +</p> +<p> +<s xml:id="echoid-s247" xml:space="preserve"> +5 ordo. +<lb/>[<emph style="it">tr: +order 5 +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s248" xml:space="preserve"> +recte +<lb/>[<emph style="it">tr: +correct +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f087v" o="87v" n="174"/> +<pb file="add_6782_f088" o="88" n="175"/> +<div xml:id="echoid-div58" type="page_commentary" level="2" n="58"> +<p> +<s xml:id="echoid-s249" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s249" xml:space="preserve"> +Harriot lists his orders 9 to 12 of Pythagorean triples, +continuing in each case until the hypotenuse is greater than 1105. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s251" xml:space="preserve"> +4) +</s> +</p> +<pb file="add_6782_f088v" o="88v" n="176"/> +<pb file="add_6782_f089" o="89" n="177"/> +<div xml:id="echoid-div59" type="page_commentary" level="2" n="59"> +<p> +<s xml:id="echoid-s252" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s252" xml:space="preserve"> +Harriot lists his orders 13 to 22 of Pythagorean triples, +continuing in each case until the hypotenuse is equal to or greater than 1105. +Unfortunately an error in the last step of order 19 has led him to miss one of the triples ending in 1105: +the final set of differences should have been 38, 84, 84, +leading to the triple (817, 744, 1105). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s254" xml:space="preserve"> +<emph style="super">5</emph>.) +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s255" xml:space="preserve"> +&c. in infinitum. +<lb/>[<emph style="it">tr: +etc. indefinitely. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f089v" o="89v" n="178"/> +<pb file="add_6782_f090" o="90" n="179"/> +<div xml:id="echoid-div60" type="page_commentary" level="2" n="60"> +<p> +<s xml:id="echoid-s256" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s256" xml:space="preserve"> +On this folio, Harriot demonstrates both geometrically and arithmetically that +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mrow><mn>8</mn><mn>5</mn></mrow><mn>2</mn></msup></mrow><mo>=</mo><mrow><msup><mrow><mn>8</mn><mn>4</mn></mrow><mn>2</mn></msup></mrow><mo>+</mo><mrow><msup><mrow><mn>1</mn><mn>2</mn></mrow><mn>2</mn></msup></mrow><mo>+</mo><mrow><msup><mn>4</mn><mn>2</mn></msup></mrow><mo>+</mo><mrow><msup><mn>3</mn><mn>2</mn></msup></mrow></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p> +<s xml:id="echoid-s258" xml:space="preserve"> +one square æquall to many. <lb/> +whence <lb/> +To devide one square into <lb/> +many +</s> +</p> +<pb file="add_6782_f090v" o="90v" n="180"/> +<pb file="add_6782_f091" o="91" n="181"/> +<div xml:id="echoid-div61" type="page_commentary" level="2" n="61"> +<p> +<s xml:id="echoid-s259" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s259" xml:space="preserve"> +An investigation into Pythagorean triples with hypotenuse 1105. +Harriot obtains several such triples by multiplication of triples already known. +Those marked 'supra +' or '+ supra' duplicate others earlier in the list. <lb/> +Three triples are identified by Harriot as prime, that is, with no common factors, +that is, (47, 1104, 1105), (264, 1073, 1105), (576, 943, 1105). +An error in his 19th order, on Add MS 6782, f. 89, has led him to miss a fourth, (817, 744, 1105). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s261" xml:space="preserve"> +supra + +<lb/>[<emph style="it">tr: +as above +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s262" xml:space="preserve"> +primi +<lb/>[<emph style="it">tr: +prime +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f091v" o="91v" n="182"/> +<pb file="add_6782_f092" o="92" n="183"/> +<pb file="add_6782_f092v" o="92v" n="184"/> +<pb file="add_6782_f093" o="93" n="185"/> +<pb file="add_6782_f093v" o="93v" n="186"/> +<pb file="add_6782_f094" o="94" n="187"/> +<pb file="add_6782_f094v" o="94v" n="188"/> +<pb file="add_6782_f095" o="95" n="189"/> +<pb file="add_6782_f095v" o="95v" n="190"/> +<div xml:id="echoid-div62" type="page_commentary" level="2" n="62"> +<p> +<s xml:id="echoid-s263" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s263" xml:space="preserve"> +Successive halving of 90 degrees. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f096" o="96" n="191"/> +<div xml:id="echoid-div63" type="page_commentary" level="2" n="63"> +<p> +<s xml:id="echoid-s265" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s265" xml:space="preserve"> +Successive halving of 90 degrees, continued from the previous page (f. 95v). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f096v" o="96v" n="192"/> +<pb file="add_6782_f097" o="97" n="193"/> +<pb file="add_6782_f097v" o="97v" n="194"/> +<pb file="add_6782_f098" o="98" n="195"/> +<pb file="add_6782_f098v" o="98v" n="196"/> +<pb file="add_6782_f099" o="99" n="197"/> +<pb file="add_6782_f099v" o="99v" n="198"/> +<pb file="add_6782_f100" o="100" n="199"/> +<pb file="add_6782_f100v" o="100v" n="200"/> +<pb file="add_6782_f101" o="101" n="201"/> +<pb file="add_6782_f101v" o="101v" n="202"/> +<div xml:id="echoid-div64" type="page_commentary" level="2" n="64"> +<p> +<s xml:id="echoid-s267" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s267" xml:space="preserve"> +A double-page calculation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>2</mn></mrow></msqrt></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f102" o="102" n="203"/> +<pb file="add_6782_f102v" o="102v" n="204"/> +<pb file="add_6782_f103" o="103" n="205"/> +<pb file="add_6782_f103v" o="103v" n="206"/> +<pb file="add_6782_f104" o="104" n="207"/> +<pb file="add_6782_f104v" o="104v" n="208"/> +<div xml:id="echoid-div65" type="page_commentary" level="2" n="65"> +<p> +<s xml:id="echoid-s269" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s269" xml:space="preserve"> +A calculation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>2</mn></mrow></msqrt></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f105" o="105" n="209"/> +<pb file="add_6782_f105v" o="105v" n="210"/> +<pb file="add_6782_f106" o="106" n="211"/> +<div xml:id="echoid-div66" type="page_commentary" level="2" n="66"> +<p> +<s xml:id="echoid-s271" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s271" xml:space="preserve"> +64 digits of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>2</mn></mrow></msqrt></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f106v" o="106v" n="212"/> +<div xml:id="echoid-div67" type="page_commentary" level="2" n="67"> +<p> +<s xml:id="echoid-s273" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s273" xml:space="preserve"> +A check on the first few digits of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>2</mn></mrow></msqrt></mstyle></math> by multiplication. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f107" o="107" n="213"/> +<div xml:id="echoid-div68" type="page_commentary" level="2" n="68"> +<p> +<s xml:id="echoid-s275" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s275" xml:space="preserve"> +A half sheet, very worn, and darker in colour than the pages that follow, +with a title and Harriot's initials. <lb/> +For a detailed account of the history and the mathematics of the treatise that follows, +see Janet Beery and Jacqueline Stedall, +<emph style="it">Thomas Harriot 19s doctrine of triangular numbers: the 'Magisteria magna'</emph> (2009). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s277" xml:space="preserve"> +De Numeris Triangularibus <lb/> +et inde <lb/> +De progressionibus Arithmeticis <lb/> +Magisteria magna <lb/> +T. H. +<lb/>[<emph style="it">tr: +On triangular numbers and thence artihmetic progressions <lb/> +The great doctrine of Thomas Harriot. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f107v" o="107v" n="214"/> +<pb file="add_6782_f108" o="108" n="215"/> +<div xml:id="echoid-div69" type="page_commentary" level="2" n="69"> +<p> +<s xml:id="echoid-s278" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s278" xml:space="preserve"> +Throughout the treatise that follows, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> is to be read as a positive integer. <lb/> +The table at the top of the page is an array of general triangular numbers: +the first row and first column contain units, the second row and second column contain lengths, +the third row and third column contain triangular numbers, the fourth row and fourth column contain pyramidal numbers, +and so on. +Between the numbers are signs that combine 'plus' and 'equals', +illustrating the additive property of the table. <lb/> +The numerators and denominators of the fractions are to be read as products. +Thus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>7</mn><mn>8</mn><mn>9</mn><mo>,</mo><mn>1</mn><mn>0</mn><mo>,</mo><mn>1</mn><mn>1</mn></mrow><mrow><mn>1</mn><mn>2</mn><mn>3</mn><mn>4</mn><mn>5</mn></mrow></mfrac></mstyle></math> in the bottom right-hand corner, for example, is to be read as +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>7</mn><mo>×</mo><mn>8</mn><mo>×</mo><mn>9</mn><mo>×</mo><mn>1</mn><mn>0</mn><mo>×</mo><mn>1</mn><mn>1</mn></mrow><mrow><mn>1</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>4</mn><mo>×</mo><mn>5</mn></mrow></mfrac><mo>=</mo><mn>9</mn><mn>2</mn><mn>4</mn></mstyle></math>. <lb/> +Below the tables Harriot has written general formulae for the numbers in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th row. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head21" xml:space="preserve"> +1.) +</head> +<pb file="add_6782_f108v" o="108v" n="216"/> +<pb file="add_6782_f109" o="109" n="217"/> +<div xml:id="echoid-div70" type="page_commentary" level="2" n="70"> +<p> +<s xml:id="echoid-s280" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s280" xml:space="preserve"> +The formulae from page 1 (Add MS 6782, f. 108) are expanded by long multiplication, +with each formula used as a starting point for the next. +The formula in the second box, for example, is obtained from the formula in the first box, +by multiplying by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo maxsize="1">)</mo></mstyle></math> and dividing by 3 (as also instructed by Cardano, see Add MS 6782, f. 44). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head22" xml:space="preserve"> +2.) +</head> +<pb file="add_6782_f109v" o="109v" n="218"/> +<pb file="add_6782_f110" o="110" n="219"/> +<div xml:id="echoid-div71" type="page_commentary" level="2" n="71"> +<p> +<s xml:id="echoid-s282" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s282" xml:space="preserve"> +In the table at the top of the page, the triangular numbers from page 1 (Add MS 6782, f. 108) +are rearranged into a triangular pattern, with the sum of each row on the right. <lb/> +In the next table, each entry from the top table is written as a fraction, as on page 1. <lb/> +Below the table are general formulae for the entries in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo maxsize="1">)</mo></mstyle></math>th row. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head23" xml:space="preserve"> +3.) +</head> +<pb file="add_6782_f110v" o="110v" n="220"/> +<pb file="add_6782_f111" o="111" n="221"/> +<div xml:id="echoid-div72" type="page_commentary" level="2" n="72"> +<p> +<s xml:id="echoid-s284" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s284" xml:space="preserve"> +The formulae from the bottom of page 3 (Add MS 6782, f. 110), expanded by long multiplication. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head24" xml:space="preserve"> +4.) +</head> +<pb file="add_6782_f111v" o="111v" n="222"/> +<pb file="add_6782_f112" o="112" n="223"/> +<div xml:id="echoid-div73" type="page_commentary" level="2" n="73"> +<p> +<s xml:id="echoid-s286" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s286" xml:space="preserve"> +At the top of the page are two differences tables. +In each case column headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> may be taken as the starting column. +Column <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> contains successive differences between entries in column <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, and so on. +As elsewhere, a triangle broadening downwards thus, Δ, indicates an increasing column. +A small square is used to indicate columns of equal entries. <lb/> +The central table shows a difference table generated from a constant differenc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. +Now the lower case letters <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> represent the first entry of each column +(the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, which for Harriot represented an un unknown quantity, is omitted). +Harriot has drawn a diagonal line under the table generated from a single entry +in the constant difference column. +Two small inset charts to the right of the main table show +the pattern of increasing and decreasing rows (here they are all increasing) +and the pattern of signs in each column (here they are all <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo></mstyle></math>). <lb/> +The lower table contains general formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo maxsize="1">)</mo></mstyle></math>th entry +in each column of the difference table, +using the triangular number coefficients established on page 3 (Add MS 6782, f. 110). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head25" xml:space="preserve"> +5.) +</head> +<pb file="add_6782_f112v" o="112v" n="224"/> +<pb file="add_6782_f113" o="113" n="225"/> +<div xml:id="echoid-div74" type="page_commentary" level="2" n="74"> +<p> +<s xml:id="echoid-s288" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s288" xml:space="preserve"> +As on page 5 (Add MS 6782, f. 112), this page begins with two difference tables, +but now the columns headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> are decreasing, indicated by a triangle narrowing downwards. <lb/> +As on page 5, the central table contains formulae for the individual entries. +The two small inset charts to the right of the main table show +the pattern of increasing and decreasing rows (here they are alternately increasing and decreasing) +and the pattern of signs in each column. <lb/> +The lower table contains general formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo maxsize="1">)</mo></mstyle></math>th entry +in each column of the difference table, +using the triangular number coefficients established on page 3 (Add MS 6782, f. 110). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head26" xml:space="preserve"> +6.) +</head> +<pb file="add_6782_f113v" o="113v" n="226"/> +<pb file="add_6782_f114" o="114" n="227"/> +<div xml:id="echoid-div75" type="page_commentary" level="2" n="75"> +<p> +<s xml:id="echoid-s290" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s290" xml:space="preserve"> +Page 7 is similar to page 6 (Add MS 6782, f. 113), +except for a change in the pattern of increasing and decreasing columns. <lb/> +The difference table at the top right contains a rare error: the last entry of column <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> +should be 1031 not 1030. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head27" xml:space="preserve"> +7.) +</head> +<pb file="add_6782_f114v" o="114v" n="228"/> +<pb file="add_6782_f115" o="115" n="229"/> +<div xml:id="echoid-div76" type="page_commentary" level="2" n="76"> +<p> +<s xml:id="echoid-s292" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s292" xml:space="preserve"> +Here Harriot has listed all possible patterns of increasing (c) and decreasing (d) columns +for difference tables of up to six columns. <lb/> +In the lower half of the page, Harriot has produced 32 charts, like those on pages 5 to 7 +(Add MS 6782, f. 111 to f. 113), showing the sign patterns in the column entries, +for each pattern of increasing and decreasing columns, for up to six columns. <lb/> +The symbols above charts 1 and 32, which look rather like <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>σ</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math>, +are Harriot's symbols for tangents and secants. +The symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> above tables 11 and 22 is his symbol for sines. In each case, +the patterns of c and d columns are those required for the corresponding trigonometric tables. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head28" xml:space="preserve"> +8.) +</head> +<pb file="add_6782_f115v" o="115v" n="230"/> +<pb file="add_6782_f116" o="116" n="231"/> +<div xml:id="echoid-div77" type="page_commentary" level="2" n="77"> +<p> +<s xml:id="echoid-s294" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s294" xml:space="preserve"> +This page shows general entries in a difference table with six columns, +generated from 24 entries in the constant difference column. +As on page 5 (Add MS 6782, f. 112), a diagonal line has been drawn below the entries +generated from just one entry in the constant difference column. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head29" xml:space="preserve"> +9.) +</head> +<pb file="add_6782_f116v" o="116v" n="232"/> +<pb file="add_6782_f117" o="117" n="233"/> +<div xml:id="echoid-div78" type="page_commentary" level="2" n="78"> +<p> +<s xml:id="echoid-s296" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s296" xml:space="preserve"> +The left hand side of the page contains a difference table with increasing columns, +generated from a constant difference 2. <lb/> +In the first table on the right, the column headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> +contains every third entry (denoted by the note <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>=</mo><mn>3</mn></mstyle></math>) from the column headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>. <lb/> +In the second table on the right, the column headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> +contains every second entry (denoted by the note <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>=</mo><mn>2</mn></mstyle></math>) from the column headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. <lb/> +The other three tables on the right are constructed in a similar way. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head30" xml:space="preserve"> +10.) +</head> +<pb file="add_6782_f117v" o="117v" n="234"/> +<pb file="add_6782_f118" o="118" n="235"/> +<div xml:id="echoid-div79" type="page_commentary" level="2" n="79"> +<p> +<s xml:id="echoid-s298" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s298" xml:space="preserve"> +This folio is very similar to the previous one (Add MS 6782, f. 117). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head31" xml:space="preserve"> +11.) +</head> +<pb file="add_6782_f118v" o="118v" n="236"/> +<pb file="add_6782_f119" o="119" n="237"/> +<div xml:id="echoid-div80" type="page_commentary" level="2" n="80"> +<p> +<s xml:id="echoid-s300" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s300" xml:space="preserve"> +This folio is similar to the previous two (Add MS 6782, f. 117 and f. 118) +except for a different pattern of increasing and decreasing columns. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head32" xml:space="preserve"> +12.) +</head> +<pb file="add_6782_f119v" o="119v" n="238"/> +<pb file="add_6782_f120" o="120" n="239"/> +<div xml:id="echoid-div81" type="page_commentary" level="2" n="81"> +<p> +<s xml:id="echoid-s302" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s302" xml:space="preserve"> +This folio is similar to the previous one (Add MS 6782, f. 119) +except for the opposite pattern of increasing and decreasing columns. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head33" xml:space="preserve"> +13.) +</head> +<pb file="add_6782_f120v" o="120v" n="240"/> +<pb file="add_6782_f121" o="121" n="241"/> +<div xml:id="echoid-div82" type="page_commentary" level="2" n="82"> +<p> +<s xml:id="echoid-s304" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s304" xml:space="preserve"> +On the right hand side of the page are three partial difference tables from earlier pages; +the first and third tables are from page 10 (Add MS 6782, f. 117) while the second is from page 11 (f. 121). +In each case, Harriot has also listed the first entries of the columns of the original difference table, +denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. <lb/> +On the left hand side are algebraic versions of the same partial difference tables. +The first table gives formulae for successive second entries +of the column beginning with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, together with successive differences. <lb/> +The second table gives formulae for successive third entries +of the column beginning with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, together with successive differences. +The third table gives formulae for succesive <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries +of the column beginning with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. <lb/> +The final table shows the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th-entry formulae multiplied out, +with differences calculated between them. +This demonstrates that the constant difference, +in a table constructed from every <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entry of the original table, is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>n</mi><mi>n</mi><mi>a</mi></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head34" xml:space="preserve"> +14.) +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s306" xml:space="preserve"> +hoc est: +<lb/>[<emph style="it">tr: +that is: +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f121v" o="121v" n="242"/> +<pb file="add_6782_f122" o="122" n="243"/> +<div xml:id="echoid-div83" type="page_commentary" level="2" n="83"> +<p> +<s xml:id="echoid-s307" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s307" xml:space="preserve"> +Following from the calculations in the previous folio (Add MS 6782, f. 121) +for the column beginning with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, +this folio gives the formulae for successive <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries +from the columns beginning with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> and with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head35" xml:space="preserve"> +15.) +</head> +<pb file="add_6782_f122v" o="122v" n="244"/> +<pb file="add_6782_f123" o="123" n="245"/> +<head xml:id="echoid-head36" xml:space="preserve"> +16.) +</head> +<div xml:id="echoid-div84" type="page_commentary" level="2" n="84"> +<p> +<s xml:id="echoid-s309" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s309" xml:space="preserve"> +Following from the calculations in the previous folios (Add MS 6782, f. 122) +for the columns beginning with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, +this folio gives the formulae for successive <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries +from the columns beginning with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f123v" o="123v" n="246"/> +<pb file="add_6782_f124" o="124" n="247"/> +<div xml:id="echoid-div85" type="page_commentary" level="2" n="85"> +<p> +<s xml:id="echoid-s311" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s311" xml:space="preserve"> +This folio shows the formulae from page 15 (Add MS 6782, f. 122) for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> column, +multiplied out in full and with differences calculated bewteen all entries. +The table is too wide to fit on the page, +so the differences appear in the boxes below the main entries. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head37" xml:space="preserve"> +17.) +</head> +<pb file="add_6782_f124v" o="124v" n="248"/> +<pb file="add_6782_f125" o="125" n="249"/> +<div xml:id="echoid-div86" type="page_commentary" level="2" n="86"> +<p> +<s xml:id="echoid-s313" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s313" xml:space="preserve"> +This folio shows the formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> columns +from pages 15 and 16 (Add MS 6782, f. 122 and f. 123) +multiplied out in full and with differences calculated between all entries. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head38" xml:space="preserve"> +18.) +</head> +<pb file="add_6782_f125v" o="125v" n="250"/> +<pb file="add_6782_f126" o="126" n="251"/> +<div xml:id="echoid-div87" type="page_commentary" level="2" n="87"> +<p> +<s xml:id="echoid-s315" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s315" xml:space="preserve"> +On this folio and the next (Add MS 6782, f. 126 and f. 127), +Harriot has used the formulae from page 17 (Add MS 6782, f. 124) +to write equations for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> +in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. +He has solved first for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>, +then for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>, +then for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>, and so on. +He has ended each calculation with 'RE', indicating 'recto' or 'correct'. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head39" xml:space="preserve"> +19.) +</head> +<p> +<s xml:id="echoid-s317" xml:space="preserve"> +1. Canon. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f126v" o="126v" n="252"/> +<pb file="add_6782_f127" o="127" n="253"/> +<div xml:id="echoid-div88" type="page_commentary" level="2" n="88"> +<p> +<s xml:id="echoid-s318" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s318" xml:space="preserve"> +This folios is the continuation of page 19 (Add MS 6782, f. 126). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head40" xml:space="preserve"> +20.) +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s320" xml:space="preserve"> +Residuum : 1. canonis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Remainder of the canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f127v" o="127v" n="254"/> +<pb file="add_6782_f128" o="128" n="255"/> +<div xml:id="echoid-div89" type="page_commentary" level="2" n="89"> +<p> +<s xml:id="echoid-s321" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s321" xml:space="preserve"> +On this folio Harriot has carried out calculations similar to those on pages 19 and 20 +(Add MS 6782, f. 126 and f. 127), +this time to find equations for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> +in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math>. <lb/> +Note that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>8</mn><mi>n</mi><mi>n</mi></mstyle></math> in line 7 should be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>8</mn><mi>n</mi><mi>n</mi><mi>n</mi></mstyle></math>; the error is corrected in the next line. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head41" xml:space="preserve"> +21.) +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s323" xml:space="preserve"> +1. Canon. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f128v" o="128v" n="256"/> +<pb file="add_6782_f129" o="129" n="257"/> +<div xml:id="echoid-div90" type="page_commentary" level="2" n="90"> +<p> +<s xml:id="echoid-s324" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s324" xml:space="preserve"> +On this folio Harriot has carried out calculations similar to those on pages 19, 20, and 21 +(Add MS 6782, f. 126, f. 127, and f. 128), +this time to find equations for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> +in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>. <lb/> +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head42" xml:space="preserve"> +22.) +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s326" xml:space="preserve"> +1. Canon. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s327" xml:space="preserve"> +1. Canon. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s328" xml:space="preserve"> +1. Canon. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f129v" o="129v" n="258"/> +<pb file="add_6782_f130" o="130" n="259"/> +<div xml:id="echoid-div91" type="page_commentary" level="2" n="91"> +<p> +<s xml:id="echoid-s329" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s329" xml:space="preserve"> +On this folio, Harriot has summarized his results from pages 19 and 20 (Add MS 6782, f. 126 and f. 127) +and has extended them to other patterns of increasing and decreasing columns. <lb/> +The symbols <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>σ</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math> in the upper tables indicate that these formulae +may be used to interpolate values in tables of tangents and secants. <lb/> +The symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> in the lower tables indicate that these formulae +may be used to interpolate tables of sines. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head43" xml:space="preserve"> +23.) +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s331" xml:space="preserve"> +Canones, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Canons for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f130v" o="130v" n="260"/> +<pb file="add_6782_f131" o="131" n="261"/> +<div xml:id="echoid-div92" type="page_commentary" level="2" n="92"> +<p> +<s xml:id="echoid-s332" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s332" xml:space="preserve"> +On this folio, Harriot has summarized his results from page 21 (Add MS 6782, f. 128) +and has extended them to other patterns of increasing and decreasing columns. <lb/> +The symbols <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>σ</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math> in the upper tables indicate that these formulae +may be used to interpolate values in tables of tangents and secants. <lb/> +The symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> in the lower tables indicate that these formulae +may be used to interpolate tables of sines. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head44" xml:space="preserve"> +24.) +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s334" xml:space="preserve"> +Canones, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Canons for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f131v" o="131v" n="262"/> +<pb file="add_6782_f132" o="132" n="263"/> +<div xml:id="echoid-div93" type="page_commentary" level="2" n="93"> +<p> +<s xml:id="echoid-s335" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s335" xml:space="preserve"> +On this folio, Harriot has summarized his results from page 22 (Add MS 6782, f. 129) +and has extended them to other patterns of increasing and decreasing columns. <lb/> +The symbols <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>σ</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math> in the upper tables indicate that these formulae +may be used to interpolate tables of tangents and secants. <lb/> +The symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> in the lower tables indicate that these formulae +may be used to interpolate tables of sines. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head45" xml:space="preserve"> +25.) +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s337" xml:space="preserve"> +Canones, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Canons for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s338" xml:space="preserve"> +Canones, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Canons for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f132v" o="132v" n="264"/> +<pb file="add_6782_f133" o="133" n="265"/> +<div xml:id="echoid-div94" type="page_commentary" level="2" n="94"> +<p> +<s xml:id="echoid-s339" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s339" xml:space="preserve"> +This folio contains a set of formulae for interpolating <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo maxsize="1">)</mo></mstyle></math> new terms between each pair of entries +in the fourth column of a difference table (the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> column). +For reasons of space, Harriot has written the entries for the fourth column below those +for the first, second, and third columns. <lb/> +At the bottom of the page, he has written a single general formula for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math>th interpolated term +in the fourth column. This he calls the 'magisterium', which may here be translated as 'rule'. +Since this formula is expressed entirely in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>, +he longer needs to compute <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head46" xml:space="preserve"> +26.) +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s341" xml:space="preserve"> +Pro Magisterio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +For the rule for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s342" xml:space="preserve"> +Magisterium. +<lb/>[<emph style="it">tr: +Rule +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f133v" o="133v" n="266"/> +<pb file="add_6782_f134" o="134" n="267"/> +<div xml:id="echoid-div95" type="page_commentary" level="2" n="95"> +<p> +<s xml:id="echoid-s343" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s343" xml:space="preserve"> +This page shows formulae analogous to the 'magisterium' on page 26 (Add MS 6782, f. 133) +for the interpolation of difference tables of up to six columns, with all columns increasing. <lb/> +The symbols <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>σ</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math> indicate that these formulae +may be used to interpolate tables of tangents and secants. <lb/> +The symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math> in this context is not a fraction, +but indicates that the expression to the left of it +is the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math>th entry of a table interpolated to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> times its original length. +The sign )=( may therefore be read as 'indexed by'. +On BL Add MS 6787, f. 352, Harriot experiments with various alternatives for the symbol )=(. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head47" xml:space="preserve"> +27.) +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s345" xml:space="preserve"> +Magisteria +<lb/>[<emph style="it">tr: +Rules +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f134v" o="134v" n="268"/> +<pb file="add_6782_f135" o="135" n="269"/> +<head xml:id="echoid-head48" xml:space="preserve"> +28.) +</head> +<div xml:id="echoid-div96" type="page_commentary" level="2" n="96"> +<p> +<s xml:id="echoid-s346" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s346" xml:space="preserve"> +Interpolation formulae as on page 27 (Add MS 6782, f. 134) but now with all columns decreasing. <lb/> +The symbols <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>σ</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math> indicate that these formulae +may be used to interpolate tables of tangents and secants. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s348" xml:space="preserve"> +Magisteria +<lb/>[<emph style="it">tr: +Rules +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f135v" o="135v" n="270"/> +<pb file="add_6782_f136" o="136" n="271"/> +<div xml:id="echoid-div97" type="page_commentary" level="2" n="97"> +<p> +<s xml:id="echoid-s349" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s349" xml:space="preserve"> +Interpolation formulae as on page 27 (Add MS 6782, f. 134) but now +for columns that are alternately increasing and decreasing. <lb/> +The symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> indicates that these formulae may be used to interpolate tables of sines. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head49" xml:space="preserve"> +29.) +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s351" xml:space="preserve"> +Magisteria +<lb/>[<emph style="it">tr: +Rules +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f136v" o="136v" n="272"/> +<pb file="add_6782_f137" o="137" n="273"/> +<div xml:id="echoid-div98" type="page_commentary" level="2" n="98"> +<p> +<s xml:id="echoid-s352" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s352" xml:space="preserve"> +Interpolation formulae as on page 29 (Add MS 6782, f. 136) but now +for columns that are alternately decreasing and increasing. <lb/> +The symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> indicates that these formulae may be used to interpolate tables of sines. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head50" xml:space="preserve"> +30.) +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s354" xml:space="preserve"> +Magisteria +<lb/>[<emph style="it">tr: +Rules +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f137v" o="137v" n="274"/> +<pb file="add_6782_f138" o="138" n="275"/> +<div xml:id="echoid-div99" type="page_commentary" level="2" n="99"> +<p> +<s xml:id="echoid-s355" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s355" xml:space="preserve"> +The numbers 4, 3, 2, and 1 in the upper righthand corners of pages 31, 32, 33, and 34, +(Add MS 6782, f. 138 to f. 141) indicate that these four folios are closely related +and could be read in either direction. <lb/> +This folio brings together the formulae from pages 27 to 30 (Add MS 6782, f. 134 to f. 137). <lb/> +The coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> +are now written as descending rather than ascending powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math>. +This means that the signs of the coefficients now follow the patterns given in the sign charts +for all increasing, or all decreasing, or alternately increasing and decreasing columns.<lb/> +The symbols <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>σ</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math> below the upper table indicate that these patterns +are required for interpolating tables of tangents and secants. <lb/> +The symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> below the lower table indicates that these patterns +are required for interpolating tables of sines. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head51" xml:space="preserve"> +31.) +</head> +<p> +<s xml:id="echoid-s357" xml:space="preserve"> +(4. +</s> +</p> +<pb file="add_6782_f138v" o="138v" n="276"/> +<pb file="add_6782_f139" o="139" n="277"/> +<div xml:id="echoid-div100" type="page_commentary" level="2" n="100"> +<p> +<s xml:id="echoid-s358" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s358" xml:space="preserve"> +This folio contains the same interpolation formulae as on page 31, +but now the coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> have been factorized. <lb/> +The inclusion of columns headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> in the inset charts, on this and the two following pages +(Add MS 6782, f. 140 and f. 141), emphasizes that the formulae may be generalized +to any number of columns. <lb/> +The symbols <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>σ</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math> below the lower left table indicate that these patterns +are required for interpolating tables of tangents and secants. <lb/> +The symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> below the lower right table indicates that these patterns +are required for interpolating tables of sines. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head52" xml:space="preserve"> +32.) +</head> +<p> +<s xml:id="echoid-s360" xml:space="preserve"> +(3. +</s> +</p> +<pb file="add_6782_f139v" o="139v" n="278"/> +<pb file="add_6782_f140" o="140" n="279"/> +<div xml:id="echoid-div101" type="page_commentary" level="2" n="101"> +<p> +<s xml:id="echoid-s361" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s361" xml:space="preserve"> +This folio contains another version of the interpolation formulae on pages 31 and 32 +(Add MS 6782, f. 138 and f. 139), +but now each coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> has its own denominator. +This appears to be Harriot's preferred form. <lb/> +The symbols <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>σ</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math> below the lower left table indicate that these patterns +are required for interpolating tables of tangents and secants. <lb/> +The symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> below the lower right table indicates that these patterns +are required for interpolating tables of sines. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head53" xml:space="preserve"> +33.) +</head> +<p> +<s xml:id="echoid-s363" xml:space="preserve"> +(2. +</s> +</p> +<pb file="add_6782_f140v" o="140v" n="280"/> +<pb file="add_6782_f141" o="141" n="281"/> +<div xml:id="echoid-div102" type="page_commentary" level="2" n="102"> +<p> +<s xml:id="echoid-s364" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s364" xml:space="preserve"> +This folio contains the same formulae as page 33 (Add MS 6782, f. 140) +but with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math> now replaced by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math>. +That is, there is no interpolation. <lb/> +This page is numbered 1 in its upper righthand corner, +making it the first in the subsequence of pages 34, 33, 32, and 31 (Add MS f. 141 to f. 138). +Harriot explained on the next page (Add MS 6782, f. 142), which is a second version of this one, +also numbered 1 in its upper righthand corner, +that one may begin with the formulae on page 34 (Add MS 6782, 141), +then derive the formulae on page 33 (Add MS 6782, f. 140) by replacing <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math> by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math>. +That is, one can reverse the sequence of pages 31 to 34. <lb/> +As on pages 32 and 33 (Add MS 6782, f. 139 and f. 140) +the sign charts show how to adapt the formulae to different patterns +of increasing and decreasing columns. <lb/> +The symbols <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>σ</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math> below the lower left table indicate that these patterns +are required for interpolating tables of tangents and secants. <lb/> +The symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> below the lower right table indicates that these patterns +are required for interpolating tables of sines. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head54" xml:space="preserve"> +34.) +</head> +<p> +<s xml:id="echoid-s366" xml:space="preserve"> +(1. +</s> +</p> +<pb file="add_6782_f141v" o="141v" n="282"/> +<pb file="add_6782_f142" o="142" n="283"/> +<div xml:id="echoid-div103" type="page_commentary" level="2" n="103"> +<p> +<s xml:id="echoid-s367" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s367" xml:space="preserve"> +On this folio, Harriot explains the relationship between the formulae on pages 33 and 34 +(Add MS 6782, f. 140 and f. 141), and their common origin in those that appeared earlier, +on page 5 (Add MS 6782, f. 112). <lb/> +In the lower half of the page, just below the dividing line, +Harriot replaces <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math> by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math> in formula 3) from page 34, +arriving at formula 3) from page 33. +In this case, he is using <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math> to denote an ordinary fraction. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head55" xml:space="preserve"> +34.) 2<emph style="super">o</emph>.) +</head> +<p> +<s xml:id="echoid-s369" xml:space="preserve"> +1.) +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s370" xml:space="preserve"> +Etsi species quæ habentur (pag: 34.1.) ortum ducunt ex (pag: 33.2.) <lb/> +Attamen primam originem videre licet pag. 5. ubi illæ omnes <lb/> +appareat notatæ. +<lb/>[<emph style="it">tr: +Although the cases we have on page 34.1 arise from those on page 33.2, +nevertheless one may see their origins on page 5 where all of them appear in notation. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s371" xml:space="preserve"> +Utile etiam ac incundum est, considerare harum reductionum (vide versa) <lb/> +ad species in pag: 33.2. quæ huius operis sunt magisteria maxima. +<lb/>[<emph style="it">tr: +It is useful and also pleasing to consider (conversely) the reduction to the cases on page 33.2, +which are the most important rules of this work. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s372" xml:space="preserve"> +Examplum unum sufficiet. +<lb/>[<emph style="it">tr: +One example will suffice. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s373" xml:space="preserve"> +Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi><mo>=</mo><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math>. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi><mo>=</mo><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math> +</emph>]<lb/> +</s> +<s xml:id="echoid-s374" xml:space="preserve"> +Et species reducta erit: (ut pag: 33.2.) et ut sequitur: +<lb/>[<emph style="it">tr: +And the cases will be reduced (as on page 33.2) and as follows: +</emph>]<lb/> +</s> +</p> +<p> +<s xml:id="echoid-s375" xml:space="preserve"> +Fit ita: Si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi><mo>=</mo><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math> <lb/> +erit: +<lb/>[<emph style="it">tr: +Let it be done thus: <lb/> +then: +</emph>]<lb/> +</s> +</p> +<p> +<s xml:id="echoid-s376" xml:space="preserve"> +Et sic de alijs speciebus. +<lb/>[<emph style="it">tr: +And so on for other cases. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f142v" o="142v" n="284"/> +<pb file="add_6782_f143" o="143" n="285"/> +<div xml:id="echoid-div104" type="page_commentary" level="2" n="104"> +<p> +<s xml:id="echoid-s377" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s377" xml:space="preserve"> +On this and the following folio (Add MS 6782, f. 144), +Harriot gives numerical examples of his interpolation method. <lb/> +At the top of the page are formulae for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math> and for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math>, +for difference tables with two columns. +Below that are four examples of tables with two columns. <lb/> +The table on the left, with columns headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>, is interpolated +first to six, then four, then five times the number of original entries; +that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> takes the values 6, then 4, then 5. +The symbol * next to the interpolated tables marks entries from the original lefthand table. <lb/> +Below the tables, the first column of working uses the formula for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math> from the top of the page, +to obtain the entries 17, 77, and 149 in the difference table on the left. <lb/> +The second column of working uses the formula for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math> from the top of the page, +to obtain one entry in each of the three remaining difference tables. <lb/> +The third column of working presents the converse problem, +showing how to solve for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math> ), +given values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> and an entry <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. +Where the solution is not an integer, Harriot replaces <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math> by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head56" xml:space="preserve"> +35.) +</head> +<pb file="add_6782_f143v" o="143v" n="286"/> +<pb file="add_6782_f144" o="144" n="287"/> +<div xml:id="echoid-div105" type="page_commentary" level="2" n="105"> +<p> +<s xml:id="echoid-s379" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s379" xml:space="preserve"> +This folio gives further numerical examples of interpolation for tables with three columns. <lb/> +As on page 35 (Add MS 6782, f. 143), Harriot gives the appropriate <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math> formulae +at the top of the page. The formula in the first line is from page 33 (Add MS 6782, f. 140), +while the formulae on the second line are from page 32 (Add MS 6782, f. 139). <lb/> +The difference table on the left (headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>) is interpolated +to five times, and then twice, the number of original entries; +that is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> takes the values 5 and 2, respectively. <lb/> +The working in the first column illustrates the use of the formula for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math> +to calculate values in the left hand table. <lb/> +The working in the second column illustrates the use of the formula for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math> +to calculate the first new value in the second table (indexed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>5</mn></mrow></mfrac></mstyle></math>), +and the third new value in the third table (indexed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>). <lb/> +The working in the third column shows, as on page 35 (Add MS 6782, f. 143), how to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math> +given values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>, and an entry <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. +In this case the working leads to a quadratic equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math>, +which Harriot solves by completing the square. +As on page 35, the solution is not an integer, and so Harriot replaces <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math> by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head57" xml:space="preserve"> +36.) +</head> +<pb file="add_6782_f144v" o="144v" n="288"/> +<pb file="add_6782_f145" o="145" n="289"/> +<div xml:id="echoid-div106" type="page_commentary" level="2" n="106"> +<p> +<s xml:id="echoid-s381" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s381" xml:space="preserve"> +This folios contains an interpolation formula from the 'Magisteria' +but instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, ... +we now have <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>1</mn></msup></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>2</mn></msup></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>3</mn></msup></mrow></mstyle></math>, ... (these are superscripts, not powers). <lb/> +The notation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>P</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>D</mi><mn>1</mn></msup></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>D</mi><mn>2</mn></msup></mrow></mstyle></math>, ... at the top of the page is unusual; +the formula is otherwise identical to formula 5) from page 33. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f145v" o="145v" n="290"/> +<pb file="add_6782_f146" o="146" n="291"/> +<div xml:id="echoid-div107" type="page_commentary" level="2" n="107"> +<p> +<s xml:id="echoid-s383" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s383" xml:space="preserve"> +Interpolated tables, see page 35 of the 'Magisteria' (Add MS 6782, f. 143). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f146v" o="146v" n="292"/> +<div xml:id="echoid-div108" type="page_commentary" level="2" n="108"> +<p> +<s xml:id="echoid-s385" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s385" xml:space="preserve"> +This folio contains interpolation formulae similar to those on page 32 (Add MS 6782, f. 139). <lb/> +It also contains a title similar, but not identical, to that at the beginning of the 'Magisteria' +(Add MS 6782, f. 107). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s387" xml:space="preserve"> +THOMÆ HARIOTI <lb/> +Magisteria <lb/> +Numerorum Trangularium <lb/> +et inde <lb/> +Progressionum Arithmeticarum <lb/> +(veteribus et recentioribus ignota) <lb/> +incognita) +<lb/>[<emph style="it">tr: +THOMAS HARRIOT'S doctrine of triangular numbers and thence arithmetic progressions <lb/> +(unkonwn and unrecognized by ancient and more recent authors) +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f147" o="147" n="293"/> +<div xml:id="echoid-div109" type="page_commentary" level="2" n="109"> +<p> +<s xml:id="echoid-s388" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s388" xml:space="preserve"> +The two lines at the end of the page have nothing to do with the mathematics above them, +and are not apparently connected with each other. +<foreign xml:lang="lat">'Porcus per taurum sequitur vestigia ferri'</foreign> +is the first line of an epitaph engraved on the tombstone of Edmund Bunny, +rector of Bolton Percy and canon of York, +who died in February 1617/8 and was interred in York Minster: +</s> +<lb/> +<quote xml:lang="lat"> +Porcus per taurum sequitur vestigia ferri  <lb/> +Anser ovem maculat, cui potum vacca ministrat. <lb/> +Expone et redde sensum +</quote> +<lb/> +<s xml:id="echoid-s389" xml:space="preserve"> +This is an illustration of synecdoche, in which a part is referred to as the whole; +thus the pig (lard, here as used to lubricate cobbler's thread) follows the footprints of the iron (needle) +through the bull (leather); the goose (quill), to whom the cow (inkhorn) provides drink (ink), stains the sheep (skin). +(The standard version has +<foreign xml:lang="lat">variat</foreign> instead of <foreign xml:lang="lat">maculat</foreign>.) +<foreign xml:lang="lat">Expone et redde sensum</foreign> +is an instruction to the student: 'explain and translate'. +It ispossible that <foreign xml:lang="lat">porcus</foreign> is a play on Percy. +If the York epitaph was Harriot's source, it gives us a possible date of 1618 for this folio. +</s> +<lb/> +<s xml:id="echoid-s390" xml:space="preserve"> +'Bombardagladiofunhastiflammiloquentes' +(Breathing bombs, swords, death, spears, and flames) +is from a Latin translation by the 16th-century German humanist Martin Crucius of a Greek verse +(perhaps also by Crucius) consisting of compound words. +The verse appears in the preface to the <emph style="it">Opera omnnia theologica</emph> (1583) of Lambert Daneau +and in the third edition of his <emph style="it">Elenchi hæreticorum</emph> (1592). +The same line appears also in Add MS 6788, f. 50. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head58" xml:space="preserve" xml:lang="lat"> +A.1. Ad numeros triangulos <lb/> +quadratos <lb/> +pentagonos &c. <lb/> +et illorum progenies +<lb/>[<emph style="it">tr: +On triangular, square, pentagonal numbers, and their progeny +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s392" xml:space="preserve"> +Generaliter +<lb/>[<emph style="it">tr: +Generally +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s393" xml:space="preserve"> +Ad progressiones arithmeticas <lb/> +incipientes ab unitate vel quovis <lb/> +numero; quolibet etiam excessu <lb/> +progredientes. +<lb/>[<emph style="it">tr: +For arithmetic porgressions starting from one or any number; proceeding with whatever excess. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s394" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. primus numerus in progressione. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. excessus. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>. numerus loci in progressione. +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. the first number in the progression <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. the excess <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>. the number of places in the progression +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s395" xml:space="preserve"> +Quomodo istæ æqautiones <lb/> +continuentur ad infinitum <lb/> +apparet in altera charta. +<lb/>[<emph style="it">tr: +By what meanst these equations may be continued indefinitely appears in another sheet. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s396" xml:space="preserve"> +porcus per taurum sequitur vestigia ferri. <lb/> +Bombardagladiofunhastiflammiloquentes. +<lb/>[<emph style="it">tr: +The pig follows the footsteps of the iron through the bull. <lb/> +Breathing bombs, swords, death, spears, and flames. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f147v" o="147v" n="294"/> +<pb file="add_6782_f148" o="148" n="295"/> +<div xml:id="echoid-div110" type="page_commentary" level="2" n="110"> +<p> +<s xml:id="echoid-s397" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s397" xml:space="preserve"> +Rules for the numbers in six successive columns, +generated from a constant difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> in column 0. +The first entry in each column is also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. <lb/> +The expression in the first box, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi></mstyle></math>, +gives the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entry in the first column (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mi>p</mi></mstyle></math>, and so on). <lb/> +The expression in the second box, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi><mi>n</mi><mo>+</mo><mn>1</mn><mi>p</mi><mi>n</mi></mstyle></math>, divided by 2, +gives the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entry in the second column (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>0</mn><mi>p</mi></mstyle></math>, and so on). <lb/> +The expression in the third box, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi><mi>n</mi><mi>n</mi><mo>+</mo><mn>3</mn><mi>p</mi><mi>n</mi><mi>n</mi><mo>+</mo><mn>2</mn><mi>p</mi><mi>n</mi></mstyle></math>, divided by 6, +gives the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entry in the third column (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>0</mn><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>0</mn><mi>p</mi></mstyle></math>, and so on). <lb/> +The layout shows how the general term in row <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mo>+</mo><mn>1</mn></mstyle></math> +is generated from the general term in row <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math>, +by multiplying by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>+</mo><mi>k</mi></mstyle></math> and dividing by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mo>+</mo><mn>1</mn></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head59" xml:space="preserve" xml:lang="lat"> +A.1.) Ad numeros triangulos et illorum progenies. +<lb/>[<emph style="it">tr: +On triangular numbers and their progeny +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s399" xml:space="preserve"> +In Via Generali. +<lb/>[<emph style="it">tr: +In a general way +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f148v" o="148v" n="296"/> +<pb file="add_6782_f149" o="149" n="297"/> +<div xml:id="echoid-div111" type="page_commentary" level="2" n="111"> +<p> +<s xml:id="echoid-s400" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s400" xml:space="preserve"> +This folio shows how to generate the coeffcients from the previous page, f. 148. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head60" xml:space="preserve" xml:lang="lat"> +A.2. Ad æquationes numerorum figuratorum <lb/> +ut continuentur ad libitum. +<lb/>[<emph style="it">tr: +On equations of figurate numbers so that they may be continued at will +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s402" xml:space="preserve"> +In genere. +<lb/>[<emph style="it">tr: +In general +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f149v" o="149v" n="298"/> +<pb file="add_6782_f150" o="150" n="299"/> +<div xml:id="echoid-div112" type="page_commentary" level="2" n="112"> +<p> +<s xml:id="echoid-s403" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s403" xml:space="preserve"> +This folio shows calculations similar to those on f. 148, but now <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>+</mo><mi>k</mi></mstyle></math> +has been replaced by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mo>+</mo><mi>k</mi><mi>d</mi></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head61" xml:space="preserve" xml:lang="lat"> +A.3. Ad æquationes numerorum figuratorum <lb/> +ut continuentur ad libitum. +<lb/>[<emph style="it">tr: +On equations of figurate numbers so that they may be continued at will +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s405" xml:space="preserve"> +in Genere. +<lb/>[<emph style="it">tr: +in general +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f150v" o="150v" n="300"/> +<pb file="add_6782_f151" o="151" n="301"/> +<pb file="add_6782_f151v" o="151v" n="302"/> +<pb file="add_6782_f152" o="152" n="303"/> +<pb file="add_6782_f152v" o="152v" n="304"/> +<pb file="add_6782_f153" o="153" n="305"/> +<div xml:id="echoid-div113" type="page_commentary" level="2" n="113"> +<p> +<s xml:id="echoid-s406" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s406"> +In the upper left table, the key column is the one beneath the sketch of a triangular prism. +The numbers beneath the sketch are those needed to construct triangular prisms +with length equal to one side of the base: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>=</mo><mn>1</mn><mo>×</mo><mn>1</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mo>=</mo><mn>2</mn><mo>×</mo><mn>3</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>8</mn><mo>=</mo><mn>3</mn><mo>×</mo><mn>6</mn></mstyle></math>, +and so on. +These are also the pentagonal-pyramidal numbers 1, 6, 18, ... as calculated in f. 156. <lb/> +Successive sums of these numbers are shown in the column to the left headed S (for sum). <lb/> +Successive differences are shown in the columns to the right. +</s> +<lb/> +<s xml:id="echoid-s407"> +In the upper right table, the key column is the one beneath the sketch of a cube. +The numbers beneath the sketch are the cube numbers, which can also be thought of +(in keeping with the previous table) as the numbers needed to construct square prisms +with height equal to one side of the base: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>=</mo><mn>1</mn><mo>×</mo><mn>1</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>8</mn><mo>=</mo><mn>2</mn><mo>×</mo><mn>4</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>7</mn><mo>=</mo><mn>3</mn><mo>×</mo><mn>9</mn></mstyle></math>, +and so on. <lb/> +Successive sums are shown in the column to the left headed S (for sum). <lb/> +Successive differences are shown in the columns to the right. <lb/> +The column beginning 1, 7, 19, 37, ... is marked 'hexagonae equianguli 19 (equiangled hexagons); +it is possible that at first glance Harriot mistook this column for +the the hexagonal-pyramidal numbers 1, 7, 22, 50, .... +</s> +<lb/> +<s xml:id="echoid-s408"> +In the lower left table, the key column is the one beneath the sketch of a pentagonal prism. +The numbers beneath the sketch are the numbers needed to construct pentagonal prisms +with length equal to one side of the base: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>=</mo><mn>1</mn><mo>×</mo><mn>1</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>0</mn><mo>=</mo><mn>2</mn><mo>×</mo><mn>5</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mn>6</mn><mo>=</mo><mn>3</mn><mo>×</mo><mn>1</mn><mn>2</mn></mstyle></math>, +and so on. <lb/> +Successive sums are shown in the column to the left headed S (for sum). <lb/> +Successive differences are shown in the columns to the right. +</s> +<lb/> +<s xml:id="echoid-s409"> +In the lower right table, the key column is the one beneath the sketch of a hexagonal prism. +The numbers beneath the sketch are the numbers needed to construct hexagonal prisms +with length equal to one side of the base: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>=</mo><mn>1</mn><mo>×</mo><mn>1</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>2</mn><mo>=</mo><mn>2</mn><mo>×</mo><mn>6</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mn>5</mn><mo>=</mo><mn>3</mn><mo>×</mo><mn>1</mn><mn>5</mn></mstyle></math>, +and so on. <lb/> +Successive sums are shown in the column headed to the left headed S (for sum). <lb/> +Successive differences are shown in the columns to the right. +</s> +<lb/> +<s xml:id="echoid-s410" xml:space="preserve"> +The formulae at the bottom of the page are for sums of +triangular, square, pentagonal, and hexagonal prisms. +Thus the sum of the first two triangular prisms is shown in the upper left table as 1 + 6 = 7. +The same number may be calculated from the formula <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>3</mn><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>0</mn><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>9</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>2</mn><mi>a</mi></mrow><mrow><mn>2</mn><mn>4</mn></mrow></mfrac></mstyle></math> +by putting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn></mstyle></math>. <lb/> +For clues as to how Harriot found and tested these formulae, +see ARITHMETIC/Figurate numbers/Fitting polynomials to sequences. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s412" xml:space="preserve"> +columnæ [triangulari]æ. <lb/> +quæ sint pyramides [pentagones]. +<lb/>[<emph style="it">tr: +Triangular prism numbers, which are also pentagonal pyramidals +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s413" xml:space="preserve"> +columnæ tetragonæ <lb/> +seu cubi. +<lb/>[<emph style="it">tr: +Four-sided prism numbers, or cubes +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s414" xml:space="preserve"> +Hexagones æquianguli. +<lb/>[<emph style="it">tr: +Equiangular hexagonals +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s415" xml:space="preserve"> +columnæ [pentagon]æ. +<lb/>[<emph style="it">tr: +Pentagonal prisms +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s416" xml:space="preserve"> +columnæ <reg norm="hexagonae" type="abbr">Hexag</reg>. +<lb/>[<emph style="it">tr: +hexagonal prisms +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s417" xml:space="preserve"> +per Reductionem +<lb/>[<emph style="it">tr: +by reduction +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f153v" o="153v" n="306"/> +<pb file="add_6782_f154" o="154" n="307"/> +<div xml:id="echoid-div114" type="page_commentary" level="2" n="114"> +<p> +<s xml:id="echoid-s418" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s418" xml:space="preserve"> +The numerical tables on the right list, from right to left: +units, lengths, triangular numbers, triangular-pyramidal naumbers, and so on. <lb/> +The expressions on the left are general formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>th entry in each column. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head62" xml:space="preserve" xml:lang="lat"> +De numeris triangulis +<lb/>[<emph style="it">tr: +On triangular numbers +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s420" xml:space="preserve"> +(<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>) est numerus locorum <lb/> +seu radix. +<lb/>[<emph style="it">tr: +(<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>) is the number of places, or the root. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s421" xml:space="preserve"> +Quomodo istæ æquationes <lb/> +continuentur ad libitum <lb/> +apparet in Chartis. A. +<lb/>[<emph style="it">tr: +How these equations may be continued at will is shown in sheet A. +</emph>]<lb/> +<sc> +Sheet A is probably Add MS 6782, f. 237. +</sc> +</s> +</p> +<p> +<s xml:id="echoid-s422" xml:space="preserve"> +The difference of difference of <lb/> +squares is 2. <lb/> +The <reg norm="difference" type="abbr">diff</reg>. +of <reg norm="difference" type="abbr">diff</reg>. +of <reg norm="difference" type="abbr">diff</reg>: <lb/> +of cubes is 6 <lb/> +of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>: is 24. <lb/> +of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> is 12. +</s> +</p> +<p> +<s xml:id="echoid-s423" xml:space="preserve"> +Of progressions the same <lb/> +make <lb/> +Triangular <lb/> +Square <lb/> +Pentagonall <lb/> +nombers +&c. +</s> +</p> +<pb file="add_6782_f154v" o="154v" n="308"/> +<pb file="add_6782_f155" o="155" n="309"/> +<div xml:id="echoid-div115" type="page_commentary" level="2" n="115"> +<p> +<s xml:id="echoid-s424" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s424" xml:space="preserve"> +The numerical table at the top of the page lists, from right to left: +odd numbers, square numbers, square-pyramidal numbers, and so on. <lb/> +The column on the far right is headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> for root, that is, +the length of the side of each figure. <lb/> +An intercalated column shows a constant difference 2. <lb/> +The column under the short line (the side) gives the sequence 1, 3, 5, ..., +in which the differences are 2. <lb/> +The column under the square gives the square numbers 1, 4, 9, ..., +whose differences are 1, 3, 5, .... <lb/> +The column under the square-pyramid gives the square-pyramidal numbers 1, 5, 14, ..., +whose differences are 1, 4, 9, .... <lb/> +And so on. <lb/> +The expressions below the table are general formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>th entry in each column. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f155v" o="155v" n="310"/> +<pb file="add_6782_f156" o="156" n="311"/> +<div xml:id="echoid-div116" type="page_commentary" level="2" n="116"> +<p> +<s xml:id="echoid-s426" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s426" xml:space="preserve"> +The numerical table at the top of the page lists +pentagonal numbers, pentagonal-pyramidal numbers, and so on. <lb/> +The column on the far right is headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> for root, that is, +the length of the side of each figure. <lb/> +An intercalated column shows a constant difference 3. <lb/> +The column under the short line (the side) gives the sequence 1, 4, 7, ..., +in which the differences are 3. <lb/> +The column under the pentagon gives the pentagonal numbers 1, 5, 12, ..., +whose differences are 1, 4, 7, .... <lb/> +The column under the pentagonal-pyramid gives the pentagonal-pyramidal numbers 1, 6, 18, ..., +whose differences are 1, 5, 12, .... <lb/> +And so on. <lb/> +The expressions below the table are general formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>th entry in each column. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p> +<s xml:id="echoid-s428" xml:space="preserve"> +Of the rootes. +</s> +</p> +<p> +<s xml:id="echoid-s429" xml:space="preserve"> +A unit is the first [first pentagonal number]. +</s> +</p> +<p> +<s xml:id="echoid-s430" xml:space="preserve"> +Omitte the first & the summe <lb/> +of the next two, is 5. [the second pentagonal number] +</s> +</p> +<p> +<s xml:id="echoid-s431" xml:space="preserve"> +Omitte two rootes, & the summe <lb/> +of the next 3, is 12. [the third pentagonal number] +</s> +</p> +<p> +<s xml:id="echoid-s432" xml:space="preserve"> +Omitte 3 rootes, & the summe <lb/> +of the next 4, is 22. [the fourth pentagonal number] <lb/> +&c. +</s> +</p> +<pb file="add_6782_f156v" o="156v" n="312"/> +<pb file="add_6782_f157" o="157" n="313"/> +<div xml:id="echoid-div117" type="page_commentary" level="2" n="117"> +<p> +<s xml:id="echoid-s433" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s433" xml:space="preserve"> +The numerical table at the top of the page lists +hexagonal numbers, hexagonal-pyramidal numbers, and so on. <lb/> +The column on the far right is headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> for root, that is, +the length of the side of each figure. <lb/> +An intercalated column shows a constant difference 4. <lb/> +The column under the short line (the side) gives the sequence 1, 5, 9, ..., +in which the differences are 4. <lb/> +The column under the hexagon gives the hexagonal numbers 1, 6, 15, ..., +whose differences are 1, 5, 9, .... <lb/> +The column under the hexagonal-pyramid gives the hexagonal-pyramidal numbers 1, 7, 22, ..., +whose differences are 1, 6, 15, .... <lb/> +And so on. <lb/> +The expressions below the table are general formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>th entry in each column. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f157v" o="157v" n="314"/> +<pb file="add_6782_f158" o="158" n="315"/> +<div xml:id="echoid-div118" type="page_commentary" level="2" n="118"> +<p> +<s xml:id="echoid-s435" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s435" xml:space="preserve"> +The numerical table at the top of the page lists +heptagonal numbers, heptagonal-pyramidal numbers, and so on. <lb/> +The column on the far right is headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> for root, that is, +the length of the side of each figure. <lb/> +An intercalated column shows a constant difference 5. <lb/> +The column under the short line (the side) gives the sequence 1, 6, 11, ..., +in which the differences are 5. <lb/> +The column under the heptagon gives the heptagonal numbers 1, 7, 18, ..., +whose differences are 1, 6, 11, .... <lb/> +The column under the heptagonal-pyramid gives the heptagonal-pyramidal numbers 1, 8, 26, ..., +whose differences are 1, 7, 18, .... <lb/> +And so on. <lb/> +The expressions below the table are general formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>th entry in each column. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f158v" o="158v" n="316"/> +<pb file="add_6782_f159" o="159" n="317"/> +<div xml:id="echoid-div119" type="page_commentary" level="2" n="119"> +<p> +<s xml:id="echoid-s437" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s437" xml:space="preserve"> +The numerical table at the top of the page lists +octagonal numbers, octagonal-pyramidal numbers, and so on. <lb/> +The column on the far right is headed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> for root, that is, +the length of the side of each figure. <lb/> +An intercalated column shows a constant difference 6. <lb/> +The column under the short line (the side) gives the sequence 1, 7, 13, ..., +in which the differences are 6. <lb/> +The column under the octagon gives the octagonal numbers 1, 8, 21, ..., +whose differences are 1, 7, 13, .... <lb/> +The column under the octagonal-pyramid gives the octagonal-pyramidal numbers 1, 9, 30, ..., +whose differences are 1, 8, 21, .... <lb/> +And so on. <lb/> +The expressions below the table are general formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>th entry in each column. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f159v" o="159v" n="318"/> +<pb file="add_6782_f160" o="160" n="319"/> +<pb file="add_6782_f160v" o="160v" n="320"/> +<pb file="add_6782_f161" o="161" n="321"/> +<pb file="add_6782_f161v" o="161v" n="322"/> +<pb file="add_6782_f162" o="162" n="323"/> +<pb file="add_6782_f162v" o="162v" n="324"/> +<pb file="add_6782_f163" o="163" n="325"/> +<div xml:id="echoid-div120" type="page_commentary" level="2" n="120"> +<p> +<s xml:id="echoid-s439" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s439" xml:space="preserve"> +Here as on Add MS 6782, f. 330, Harriot is using <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>v</mi></mstyle></math> notation, with superscripts 0, 1, 2, 3, ... +for successive entries in a general row of the table. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head63" xml:space="preserve" xml:lang="lat"> +Elementa triangularium. +<lb/>[<emph style="it">tr: +The elements of triangular numbers +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s441" xml:space="preserve"> +Rationes compositas: et <lb/> +componentes. +<lb/>[<emph style="it">tr: +Ratios compounded, and their components. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f163v" o="163v" n="326"/> +<pb file="add_6782_f164" o="164" n="327"/> +<div xml:id="echoid-div121" type="page_commentary" level="2" n="121"> +<p> +<s xml:id="echoid-s442" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s442" xml:space="preserve"> +A table that appears to have been produced from the various rules set out on f. 163. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f164v" o="164v" n="328"/> +<pb file="add_6782_f165" o="165" n="329"/> +<div xml:id="echoid-div122" type="page_commentary" level="2" n="122"> +<p> +<s xml:id="echoid-s444" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s444" xml:space="preserve"> +Some experiments with Pascal's triangle. <lb/> +On the left, and again at the bottom of the page, the tables have been extended upwards, +giving rise to negative triangular numbers. <lb/> +On the right, the table has been multiplied throughout by 3; see also Add MS 6785, f. 83. <lb/> +Note Harriot's use of superscript notation (not powers): <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>2</mn></msup></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>3</mn></msup></mrow></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s446" xml:space="preserve"> +Male +<lb/>[<emph style="it">tr: +badly done +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s447" xml:space="preserve"> +Bene +<lb/>[<emph style="it">tr: +better +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f165v" o="165v" n="330"/> +<pb file="add_6782_f166" o="166" n="331"/> +<pb file="add_6782_f166v" o="166v" n="332"/> +<pb file="add_6782_f167" o="167" n="333"/> +<pb file="add_6782_f167v" o="167v" n="334"/> +<pb file="add_6782_f168" o="168" n="335"/> +<pb file="add_6782_f168v" o="168v" n="336"/> +<pb file="add_6782_f169" o="169" n="337"/> +<div xml:id="echoid-div123" type="page_commentary" level="2" n="123"> +<p> +<s xml:id="echoid-s448" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s448" xml:space="preserve"> +These three tables show the formulae for the entries in each column of a table generated +from a constant difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, with different patterns ofincreasing and decreasing columns. <lb/> +Increasing columns are indicated by a triangle that broadens at the bottom, thus, Δ; +decreasing columns are indicated by a trinagle that narrows at the bottom. <lb/> +Entries in the first column (after <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>) are denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>1</mn></msup></mrow></mstyle></math> +(where 1 is a superscript, not a power). <lb/> +Entries in the second column are denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>2</mn></msup></mrow></mstyle></math>. +Entries in the third column are denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>3</mn></msup></mrow></mstyle></math>. +Entries in the fourth column are denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>4</mn></msup></mrow></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head64" xml:lang="lat"> +Ad progressiones, <lb/> +<lb/>[<emph style="it">tr: +On progressions +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s450" xml:space="preserve"> +<sc> +Differences in tables of sines follow the alternately increasing and decreasing pattern given here. +</sc> +ad calculum sinuum +<lb/>[<emph style="it">tr: +for the calculation of sines +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s451" xml:space="preserve"> +omnes <reg norm="examinantur" type="abbr">exam</reg>. +<lb/>[<emph style="it">tr: +all examined +</emph>]<lb/> +</s> +</p> +<head xml:id="echoid-head65" xml:lang="lat"> +Ad progressiones, +<lb/>[<emph style="it">tr: +On progressioins +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s452" xml:space="preserve"> +<sc> +Differences in tables of sines follow the alternately increasing and decreasing pattern given here. +</sc> +ad calculum sinuum +<lb/>[<emph style="it">tr: +for the calculation of sines +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s453" xml:space="preserve"> +omnes <reg norm="examinantur" type="abbr">exam</reg>. +<lb/>[<emph style="it">tr: +all examined +</emph>]<lb/> +</s> +</p> +<head xml:id="echoid-head66" xml:lang="lat"> +Ad progressiones, +<lb/>[<emph style="it">tr: +On progressions +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s454" xml:space="preserve"> +omnes <reg norm="examinantur" type="abbr">exam</reg>. <lb/> +<lb/>[<emph style="it">tr: +all examined +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f169v" o="169v" n="338"/> +<pb file="add_6782_f170" o="170" n="339"/> +<div xml:id="echoid-div124" type="page_commentary" level="2" n="124"> +<p> +<s xml:id="echoid-s455" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s455" xml:space="preserve"> +At the top right is a numerical table in which every column is decreasing. +The working below the table demonstrates in detail how the entries are calculated. <lb/> +The table below the first double line gives formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries in each column. <lb/> +The table below the second double line gives the same information in a rearranged form. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head67" xml:lang="lat"> +Ad progressiones. +<lb/>[<emph style="it">tr: +On progressions +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s457" xml:space="preserve"> +omnes <reg norm="examinantur" type="abbr">exam</reg>. +<lb/>[<emph style="it">tr: +all examined +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f170v" o="170v" n="340"/> +<pb file="add_6782_f171" o="171" n="341"/> +<div xml:id="echoid-div125" type="page_commentary" level="2" n="125"> +<p> +<s xml:id="echoid-s458" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s458" xml:space="preserve"> +At the top right is a numerical example in which the columns alternately increase and decrease. +The working below the table demonstrates in detail how the entries are calculated. <lb/> +The first table below the double line gives formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries in each column. <lb/> +The second table gives the same information but with the fractions rearranged over common denominators. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head68" xml:lang="lat"> +Ad progressiones. +<lb/>[<emph style="it">tr: +On progressions +</emph>]<lb/> +</head> +<pb file="add_6782_f171v" o="171v" n="342"/> +<pb file="add_6782_f172" o="172" n="343"/> +<div xml:id="echoid-div126" type="page_commentary" level="2" n="126"> +<p> +<s xml:id="echoid-s460" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s460" xml:space="preserve"> +At the top of the page is a numerical example in which the columns alternately decrease and increase. +The working demonstrates in detail how the entries are calculated. <lb/> +The small table halfway down on the right contains negative entries in three of its columns. <lb/> +The table at the bottom of the page gives formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries in each column. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head69" xml:lang="lat"> +Ad progressiones. +<lb/>[<emph style="it">tr: +On progressions +</emph>]<lb/> +</head> +<pb file="add_6782_f172v" o="172v" n="344"/> +<pb file="add_6782_f173" o="173" n="345"/> +<div xml:id="echoid-div127" type="page_commentary" level="2" n="127"> +<p> +<s xml:id="echoid-s462" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s462" xml:space="preserve"> +Rules for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries in the third and fourth columns of a table +generated from a constant difference, +for various patterns of increasing and decreasing columns. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f173v" o="173v" n="346"/> +<pb file="add_6782_f174" o="174" n="347"/> +<div xml:id="echoid-div128" type="page_commentary" level="2" n="128"> +<p> +<s xml:id="echoid-s464" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s464" xml:space="preserve"> +Rules for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries in the third and fourth columns of a table +generated from a constant difference, +for various patterns of increasing and decreasing columns. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f174v" o="174v" n="348"/> +<pb file="add_6782_f175" o="175" n="349"/> +<div xml:id="echoid-div129" type="page_commentary" level="2" n="129"> +<p> +<s xml:id="echoid-s466" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s466" xml:space="preserve"> +At the top right is a numerical example in which the first three columns increase but the fourth decreases. +The working below the table demonstrates in detail how the entries are calculated. <lb/> +The rule for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entry in the fourth column is given in full, in two different versions. <lb/> +Below the double line, similar rules are given for a table in which the first and second columns increase +but the third and fourth columns decrease. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f175v" o="175v" n="350"/> +<pb file="add_6782_f176" o="176" n="351"/> +<div xml:id="echoid-div130" type="page_commentary" level="2" n="130"> +<p> +<s xml:id="echoid-s468" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s468" xml:space="preserve"> +General rules for the entries in the third and fourth columns of a table +generated from a constant difference. <lb/> +Instructions for the correct signs are given in a separate note at the end of the page. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head70" xml:lang="lat"> +Ad <emph style="st">finales</emph> æquationes <lb/> +generalis methodus +<lb/>[<emph style="it">tr: +On equations, general method +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s470" xml:space="preserve"> +In <emph style="st">Descendibus</emph> <emph style="super">Decrescentibus</emph>: adde et subtrahe. +<lb/>[<emph style="it">tr: +In decreasing progressions: add and subtract. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s471" xml:space="preserve"> +In Crescentibus: subtrahe et adde. +<lb/>[<emph style="it">tr: +In increasing progressions: subtract and add. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f176v" o="176v" n="352"/> +<pb file="add_6782_f177" o="177" n="353"/> +<div xml:id="echoid-div131" type="page_commentary" level="2" n="131"> +<p> +<s xml:id="echoid-s472" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s472" xml:space="preserve"> +Rules for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries in the fifth and sixth columns of a table +generated from a constant difference, +for various patterns of increasing and decreasing columns. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f177v" o="177v" n="354"/> +<pb file="add_6782_f178" o="178" n="355"/> +<div xml:id="echoid-div132" type="page_commentary" level="2" n="132"> +<p> +<s xml:id="echoid-s474" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s474" xml:space="preserve"> +General rules for the entries in the fifth and sixth columns of a table +generated from a constant difference. <lb/> +Instructions for the correct signs are given in a separate note at the end of the page. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s476" xml:space="preserve"> +In crescentibus. subtrahe et adde. +<lb/>[<emph style="it">tr: +In increasing porgressions, subtract and add. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s477" xml:space="preserve"> +In decrescentibus. adde et subtrahe. +<lb/>[<emph style="it">tr: +In decreasing progressions, add and subtract. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f178v" o="178v" n="356"/> +<div xml:id="echoid-div133" type="page_commentary" level="2" n="133"> +<p> +<s xml:id="echoid-s478" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s478" xml:space="preserve"> +The leftmost column is marked at the top as increasing (by the triangle widening downwards. Δ), +but after a while it begins to decrease again. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head71" xml:lang="lat"> +Nota. +<lb/>[<emph style="it">tr: +Note. +</emph>]<lb/> +</head> +<pb file="add_6782_f179" o="179" n="357"/> +<pb file="add_6782_f179v" o="179v" n="358"/> +<pb file="add_6782_f180" o="180" n="359"/> +<div xml:id="echoid-div134" type="page_commentary" level="2" n="134"> +<p> +<s xml:id="echoid-s480" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s480" xml:space="preserve"> +This folio shows all combinations of 0, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, +including in each case the null combination consisting only of 0. As on Add MS 6782, f. 35, +each set of combinations is constructed from the previous one +by adding the new letter to the end of each existing combination. +This shows clearly why the number of combinations doubles with each new letter. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f180v" o="180v" n="360"/> +<pb file="add_6782_f181" o="181" n="361"/> +<div xml:id="echoid-div135" type="page_commentary" level="2" n="135"> +<p> +<s xml:id="echoid-s482" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s482" xml:space="preserve"> +Combinations of two or more quantities, generated by multiplication. <lb/> +The letters <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>, stand for +<foreign xml:lang="lat">pondus</foreign> (weight), +<foreign xml:lang="lat">magnitudo</foreign> (magnitude), +<foreign xml:lang="lat">figura</foreign> (area), +<foreign xml:lang="lat">situs</foreign> (place) +(see Add MS 6786, f. 291). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f181v" o="181v" n="362"/> +<pb file="add_6782_f182" o="182" n="363"/> +<div xml:id="echoid-div136" type="page_commentary" level="2" n="136"> +<p> +<s xml:id="echoid-s484" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s484" xml:space="preserve"> +This folio lists all the permutations of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>,<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>, +and begins lists for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi><mi>e</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi><mi>e</mi><mi>f</mi></mstyle></math>. <lb/> +The totals are written as factorials, that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mo>=</mo><mn>1</mn><mo>×</mo><mn>2</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mo>=</mo><mn>1</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>3</mn></mstyle></math>, and so on. +This enables Harriot to calculate the number of permutations of 4 or 5 letters as 120 or 720, respectively, +without writing out the entire list. <lb/> +The calculations at the bottom of the page show how each total is obtained from the previous one; +for example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>4</mn><mo>=</mo><mn>4</mn><mo>×</mo><mn>6</mn></mstyle></math>, and so on. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f182v" o="182v" n="364"/> +<pb file="add_6782_f183" o="183" n="365"/> +<div xml:id="echoid-div137" type="page_commentary" level="2" n="137"> +<p> +<s xml:id="echoid-s486" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s486" xml:space="preserve"> +The square, cube, and fourth power of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>, possibly in connection with combinations, +which is the predominant subject in the surrounding pages. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f183v" o="183v" n="366"/> +<pb file="add_6782_f184" o="184" n="367"/> +<div xml:id="echoid-div138" type="page_commentary" level="2" n="138"> +<p> +<s xml:id="echoid-s488" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s488" xml:space="preserve"> +Tables showing all possible throws of 1, 2, or 3 dice. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f184v" o="184v" n="368"/> +<pb file="add_6782_f185" o="185" n="369"/> +<div xml:id="echoid-div139" type="page_commentary" level="2" n="139"> +<p> +<s xml:id="echoid-s490" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s490" xml:space="preserve"> +This folio is a summary of Harriot's calculations on dice. <lb/> +The tables on the left, after the page has been turned sideways, +are frequency tables for the possible sums that can be obtained by throwing 1, 2, 3, 4, 5, or 6 dice +(see Add MS 6782, f. 41 and f. 40v). +The totals in each case are the appropriate powers of 6. <lb/> +The tables on the right are summaries of the tables that appear on Add MS 6782, f. 50, +and indicate the number of ways repetitions can occur. <lb/> +Amongst the calculations at the bottom right are: <lb/> +(i) a table that appears to convert hours to £, at £30 per hour. <lb/> +(ii) a conversion of 46,656 shillings (one for each possibility for throws of six dice) into £ and shillings. <lb/> +Table (i) is based on £30 or 600 shillings per hour; +converting shillings to throws of the dice, as suggested by table (ii), +gives 600 throws per hour, or 10 throws per minute. <lb/> +(iii) the ratio of throws with repetitions to throws with no repetition, for six dice, +namely (see the table above), <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mn>5</mn><mo>,</mo><mn>9</mn><mn>3</mn><mn>6</mn><mo>:</mo><mn>7</mn><mn>2</mn><mn>0</mn><mo>=</mo><mn>6</mn><mn>3</mn><mfrac><mrow><mn>5</mn><mn>7</mn><mn>6</mn></mrow><mrow><mn>7</mn><mn>2</mn><mn>0</mn></mrow></mfrac><mo>:</mo><mn>1</mn></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f185v" o="185v" n="370"/> +<pb file="add_6782_f186" o="186" n="371"/> +<pb file="add_6782_f186v" o="186v" n="372"/> +<pb file="add_6782_f187" o="187" n="373"/> +<pb file="add_6782_f187v" o="187v" n="374"/> +<pb file="add_6782_f188" o="188" n="375"/> +<pb file="add_6782_f188v" o="188v" n="376"/> +<pb file="add_6782_f189" o="189" n="377"/> +<pb file="add_6782_f189v" o="189v" n="378"/> +<pb file="add_6782_f190" o="190" n="379"/> +<pb file="add_6782_f190v" o="190v" n="380"/> +<pb file="add_6782_f191" o="191" n="381"/> +<pb file="add_6782_f191v" o="191v" n="382"/> +<pb file="add_6782_f192" o="192" n="383"/> +<pb file="add_6782_f192v" o="192v" n="384"/> +<pb file="add_6782_f193" o="193" n="385"/> +<div xml:id="echoid-div140" type="page_commentary" level="2" n="140"> +<p> +<s xml:id="echoid-s492" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s492" xml:space="preserve"> +A partial draft of page 33 of the 'Magisteria' (Add MS 6782, f. 140). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f193v" o="193v" n="386"/> +<div xml:id="echoid-div141" type="page_commentary" level="2" n="141"> +<p> +<s xml:id="echoid-s494" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s494" xml:space="preserve"> +Examples from pages 35 and 36 of the 'Magisteria' (Add MS 6782, f. 142 and f. 143). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f194" o="194" n="387"/> +<pb file="add_6782_f194v" o="194v" n="388"/> +<pb file="add_6782_f195" o="195" n="389"/> +<div xml:id="echoid-div142" type="page_commentary" level="2" n="142"> +<p> +<s xml:id="echoid-s496" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s496" xml:space="preserve"> +A draft of page 31 of the 'Magisteria' (Add MS 6782, f. 138). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head72" xml:space="preserve" xml:lang="lat"> +Magisteria. +<lb/>[<emph style="it">tr: +Rules +</emph>]<lb/> +</head> +<pb file="add_6782_f195v" o="195v" n="390"/> +<pb file="add_6782_f196" o="196" n="391"/> +<div xml:id="echoid-div143" type="page_commentary" level="2" n="143"> +<p> +<s xml:id="echoid-s498" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s498" xml:space="preserve"> +A draft of page 26 of the 'Magisteria' (Add MS 6782, f. 133). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head73" xml:space="preserve" xml:lang="lat"> +Pro Magisterio. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +For rules for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s500" xml:space="preserve"> +Magisterium. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Rule for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s501" xml:space="preserve"> +Hinc apparet quod <emph style="super">hoc</emph> +magisterium fit ex primo numero progressionis et prima <lb/> +differentia, in primo canone, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, scribendo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi><mi>N</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi><mi>N</mi><mi>N</mi></mstyle></math>. +<emph style="st">[???]</emph> grad<emph style="super">at</emph>im <lb/> +ut in exemplo. +<lb/>[<emph style="it">tr: +Here it is clear that this rule stems from the first number of the progression and the first difference, +in the first canon, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, writing <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi><mi>N</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi><mi>N</mi><mi>N</mi></mstyle></math>. +</emph>]<lb/> +</s> +<s xml:id="echoid-s502" xml:space="preserve"> +Idem observandum ex <emph style="st">cæteris</emph> +alijs canonibus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, pro <emph style="st">alijs</emph> cæteris <lb/> +magisterijs. +<lb/>[<emph style="it">tr: +The same is to be observed from other canons for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, for the remaining rules. +</emph>]<lb/> +</s> +<s xml:id="echoid-s503" xml:space="preserve"> +Similiter agendum pro alijs omnibus Magisterijs <lb/> +cæterarum progressionum. +<emph style="st">mutandis</emph> <emph style="super">notandis</emph>affectionibus secundum speciem. +<lb/>[<emph style="it">tr: +Similarly it may be done for all other rules for the remaining progressions, +noting the sign of the second case. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f196v" o="196v" n="392"/> +<head xml:id="echoid-head74" xml:space="preserve"> +Variation of ye needle <lb/> +offered <emph style="super">by</emph> Schouten in his navigation <lb/> +about ye world. +</head> +<p> +<s xml:id="echoid-s504" xml:space="preserve"> +To the southeward of the east mouth of the strayts <lb/> +of Magellan in the sight of 57.88 <lb/> +variatio. 12.0. to the NE. +</s> +</p> +<p> +<s xml:id="echoid-s505" xml:space="preserve"> +To the southward of the westmost <emph style="super">of Magellane</emph> & 20 legues more <lb/> +westward in 55.43 lat <lb/> +variatio 11.0 (NE). +</s> +</p> +<p> +<s xml:id="echoid-s506" xml:space="preserve"> +To the southern altitude 17 & 20 degrees westward <lb/> +from the middest of Magel: straytes, in the common plot <lb/> +of æquall degrees. <lb/> +variatio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> poynt or 6. NW. +</s> +</p> +<p> +<s xml:id="echoid-s507" xml:space="preserve"> +20. degrees more westward. lat: [???]: 14.12 +variatio. nulla. +</s> +</p> +<pb file="add_6782_f197" o="197" n="393"/> +<div xml:id="echoid-div144" type="page_commentary" level="2" n="144"> +<p> +<s xml:id="echoid-s508" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s508" xml:space="preserve"> +A partial draft of page 32 of the 'Magisteria' (Add MS 6782, f. 139). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f197v" o="197v" n="394"/> +<pb file="add_6782_f198" o="198" n="395"/> +<div xml:id="echoid-div145" type="page_commentary" level="2" n="145"> +<p> +<s xml:id="echoid-s510" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s510" xml:space="preserve"> +Rules for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries in four columns of a table generated from a constant difference, +as on Add MS 6782, f 169–176, but here using the notation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> (for sum) instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>v</mi></mstyle></math>, +to indicate that the entries may be thought of as sums (of lower numbered columns) +rather than differences (of higher numbered columns). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head75" xml:lang="lat"> +Ad aggregata seu summas progressionum. +<lb/>[<emph style="it">tr: +On the gathering together or sum of progressions. +</emph>]<lb/> +</head> +<pb file="add_6782_f198v" o="198v" n="396"/> +<pb file="add_6782_f199" o="199" n="397"/> +<div xml:id="echoid-div146" type="page_commentary" level="2" n="146"> +<p> +<s xml:id="echoid-s512" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s512" xml:space="preserve"> +A partial draft of page 32 of the 'Magisteria' (Add MS 6782, f. 139). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s514" xml:space="preserve"> +& In Infinitum. +<lb/>[<emph style="it">tr: +etc. Indefinitely. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f199v" o="199v" n="398"/> +<pb file="add_6782_f200" o="200" n="399"/> +<div xml:id="echoid-div147" type="page_commentary" level="2" n="147"> +<p> +<s xml:id="echoid-s515" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s515" xml:space="preserve"> +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, see page 23 of the 'Magisteria' (Add MS 6782, f. 130). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f200v" o="200v" n="400"/> +<pb file="add_6782_f201" o="201" n="401"/> +<div xml:id="echoid-div148" type="page_commentary" level="2" n="148"> +<p> +<s xml:id="echoid-s517" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s517" xml:space="preserve"> +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, see page 22 of the 'Magisteria' (Add MS 6782, f. 129). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head76" xml:space="preserve" xml:lang="lat"> +Canon <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> +<lb/>[<emph style="it">tr: +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> +</emph>]<lb/> +</head> +<pb file="add_6782_f201v" o="201v" n="402"/> +<pb file="add_6782_f202" o="202" n="403"/> +<div xml:id="echoid-div149" type="page_commentary" level="2" n="149"> +<p> +<s xml:id="echoid-s519" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s519" xml:space="preserve"> +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, see page 25 of the 'Magisteria' (Add MS 6782, f. 132). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head77" xml:space="preserve" xml:lang="lat"> +Canon <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> +</emph>]<lb/> +</head> +<pb file="add_6782_f202v" o="202v" n="404"/> +<pb file="add_6782_f203" o="203" n="405"/> +<div xml:id="echoid-div150" type="page_commentary" level="2" n="150"> +<p> +<s xml:id="echoid-s521" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s521" xml:space="preserve"> +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, see pages 22 and 25 of the 'Magisteria' (Add MS 6782, f. 129 and f. 132). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head78" xml:space="preserve" xml:lang="lat"> +De canone pro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>: +<lb/>[<emph style="it">tr: +On the canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s523" xml:space="preserve"> +Videlicet: pro 4<emph style="super">ta</emph> progressione, +3<emph style="super">m</emph> differentiarum gradualium. +<lb/>[<emph style="it">tr: +Clearly set out: for the fourth progression, three grades of differences. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s524" xml:space="preserve"> +Cuius <lb/> +species <lb/> +hic +<lb/>[<emph style="it">tr: +For these cases here +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s525" xml:space="preserve"> +Et affectio-<lb/> +nes speciem <lb/> +ita. +<lb/>[<emph style="it">tr: +And for these cases of sign thus. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s526" xml:space="preserve"> +Sed si species progressiones <lb/> +sit: +<lb/>[<emph style="it">tr: +But if the cases of progressions are: +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s527" xml:space="preserve"> +Affectiones erunt. +<lb/>[<emph style="it">tr: +The signs will be +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s528" xml:space="preserve"> +Etiam, si species <lb/> +sit: +<lb/>[<emph style="it">tr: +Also, if the cases are: +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s529" xml:space="preserve"> +Affectiones erunt. +<lb/>[<emph style="it">tr: +The signs will be +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s530" xml:space="preserve"> +Et similiter de alijs +<lb/>[<emph style="it">tr: +And similarly for others. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f203v" o="203v" n="406"/> +<pb file="add_6782_f204" o="204" n="407"/> +<head xml:id="echoid-head79" xml:space="preserve" xml:lang="lat"> +Canon. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> +</emph>]<lb/> +</head> +<pb file="add_6782_f204v" o="204v" n="408"/> +<pb file="add_6782_f205" o="205" n="409"/> +<div xml:id="echoid-div151" type="page_commentary" level="2" n="151"> +<p> +<s xml:id="echoid-s531" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s531" xml:space="preserve"> +Canons 3 and 4 for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, with columns alternately increasing and decreasing; +see page 24 of the 'Magisteria' (Add MS 6782, f. 131). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head80" xml:space="preserve" xml:lang="lat"> +3. Canon <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> +</emph>]<lb/> +</head> +<head xml:id="echoid-head81" xml:space="preserve" xml:lang="lat"> +4. Canon <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> +</emph>]<lb/> +</head> +<pb file="add_6782_f205v" o="205v" n="410"/> +<pb file="add_6782_f206" o="206" n="411"/> +<div xml:id="echoid-div152" type="page_commentary" level="2" n="152"> +<p> +<s xml:id="echoid-s533" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s533" xml:space="preserve"> +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, see page 21 of the 'Magisteria' (Add MS 6782, f. 128). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head82" xml:space="preserve" xml:lang="lat"> +De canone ad dividendam progressionum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +On the canon for dividing the progression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s535" xml:space="preserve"> +Videlicet; 5<emph style="super">tam</emph> progressionem, +4<emph style="super">or</emph> differentiarum gradualium. +<lb/>[<emph style="it">tr: +Clearly set out: the fifth progression, four grades of differences. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f206v" o="206v" n="412"/> +<pb file="add_6782_f207" o="207" n="413"/> +<div xml:id="echoid-div153" type="page_commentary" level="2" n="153"> +<p> +<s xml:id="echoid-s536" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s536" xml:space="preserve"> +Canon 1 for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, with all rows increasing; +see page 24 of the 'Magisteria' (Add MS 6782, f. 131). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head83" xml:space="preserve" xml:lang="lat"> +1. Canon <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> +</emph>]<lb/> +</head> +<pb file="add_6782_f207v" o="207v" n="414"/> +<pb file="add_6782_f208" o="208" n="415"/> +<div xml:id="echoid-div154" type="page_commentary" level="2" n="154"> +<p> +<s xml:id="echoid-s538" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s538" xml:space="preserve"> +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, see page 21 of the 'Magisteria' (Add MS 6782, f. 128). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head84" xml:space="preserve" xml:lang="lat"> +Canon <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> +</emph>]<lb/> +</head> +<pb file="add_6782_f208v" o="208v" n="416"/> +<pb file="add_6782_f209" o="209" n="417"/> +<div xml:id="echoid-div155" type="page_commentary" level="2" n="155"> +<p> +<s xml:id="echoid-s540" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s540" xml:space="preserve"> +Canon 2 for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, with all rows decreasing; +see page 24 of the 'Magisteria' (Add MS 6782, f. 131). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head85" xml:space="preserve" xml:lang="lat"> +2. Canon <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> +</emph>]<lb/> +</head> +<pb file="add_6782_f209v" o="209v" n="418"/> +<pb file="add_6782_f210" o="210" n="419"/> +<div xml:id="echoid-div156" type="page_commentary" level="2" n="156"> +<p> +<s xml:id="echoid-s542" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s542" xml:space="preserve"> +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, see pages 19–20 of the 'Magisteria' (Add MS 6782, f. 126 and f. 127). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head86" xml:space="preserve" xml:lang="lat"> +1. Canon. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> +</emph>]<lb/> +</head> +<pb file="add_6782_f210v" o="210v" n="420"/> +<div xml:id="echoid-div157" type="page_commentary" level="2" n="157"> +<p> +<s xml:id="echoid-s544" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s544" xml:space="preserve"> +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, see pages 19–20 of the 'Magisteria' (Add MS 6782, f. 126 and f. 127). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f211" o="211" n="421"/> +<div xml:id="echoid-div158" type="page_commentary" level="2" n="158"> +<p> +<s xml:id="echoid-s546" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s546" xml:space="preserve"> +Canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, see pages 19-20 of the 'Magisteria' (Add MS 6782, f. 126 and f. 127). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head87" xml:space="preserve" xml:lang="lat"> +De canone ad dividendam progressionum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +On the canon for dividing the progression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s548" xml:space="preserve"> +Videlicet; sextam progressionem, +quinque differentiarum gradualium <lb/> +gradatarum. +<lb/>[<emph style="it">tr: +Clearly set out: the sixth progression, five grades of differences. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f211v" o="211v" n="422"/> +<div xml:id="echoid-div159" type="page_commentary" level="2" n="159"> +<p> +<s xml:id="echoid-s549" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s549" xml:space="preserve"> +Sign patterns for columns of difference tables. <lb/> +For further examples see page 8 of the 'Magisteria' (Add MS 6782, f. 115). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f212" o="212" n="423"/> +<div xml:id="echoid-div160" type="page_commentary" level="2" n="160"> +<p> +<s xml:id="echoid-s551" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s551" xml:space="preserve"> +Difference tables showing some possible variations of increasing (c) and decreasing (d) columns. <lb/> +The charts on the right hand side show sign patterns for each column. <lb/> +For further examples see page 8 of the 'Magisteria' (Add MS 6782, f. 115). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f212v" o="212v" n="424"/> +<pb file="add_6782_f213" o="213" n="425"/> +<div xml:id="echoid-div161" type="page_commentary" level="2" n="161"> +<p> +<s xml:id="echoid-s553" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s553" xml:space="preserve"> +Difference tables showing some possible variations of increasing (c) and decreasing (d) columns. <lb/> +The charts on the right hand side show the sign patterns for each column. <lb/> +For further examples see page 8 of the 'Magisteria' (Add MS 6782, f. 115). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f213v" o="213v" n="426"/> +<pb file="add_6782_f214" o="214" n="427"/> +<div xml:id="echoid-div162" type="page_commentary" level="2" n="162"> +<p> +<s xml:id="echoid-s555" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s555" xml:space="preserve"> +Difference tables showing some possible variations of increasing (c) and decreasing (d) columns. <lb/> +The charts on the right hand side show the sign patterns for each column. <lb/> +For further examples see page 8 of the 'Magisteria' (Add MS 6782, f. 115). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f214v" o="214v" n="428"/> +<pb file="add_6782_f215" o="215" n="429"/> +<div xml:id="echoid-div163" type="page_commentary" level="2" n="163"> +<p> +<s xml:id="echoid-s557" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s557" xml:space="preserve"> +Lists of all possible variations of increasing (c) and decreasing (d) columns. <lb/> +See page 8 of the 'Magisteria' (Add MS 6782, f. 115). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f215v" o="215v" n="430"/> +<div xml:id="echoid-div164" type="page_commentary" level="2" n="164"> +<p> +<s xml:id="echoid-s559" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s559" xml:space="preserve"> +Lists of all possible variations of increasing (c) and decreasing (d) columns. <lb/> +The charts on the right hand side show sign patterns for individual columns. <lb/> +For a full array of such charts see page 8 of the 'Magisteria' (Add MS 6782, f. 115). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f216" o="216" n="431"/> +<div xml:id="echoid-div165" type="page_commentary" level="2" n="165"> +<p> +<s xml:id="echoid-s561" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s561" xml:space="preserve"> +Difference tables showing some possible variations of increasing (c) and decreasing (d) columns. <lb/> +The charts on the right hand side show the sign patterns for each column. <lb/> +For further examples see page 8 of the 'Magisteria' (Add MS 6782, f. 115). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f216v" o="216v" n="432"/> +<div xml:id="echoid-div166" type="page_commentary" level="2" n="166"> +<p> +<s xml:id="echoid-s563" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s563" xml:space="preserve"> +Sign patterns for columns of difference tables. <lb/> +For further examples see page 8 of the 'Magisteria' (Add MS 6782, f. 115). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f217" o="217" n="433"/> +<div xml:id="echoid-div167" type="page_commentary" level="2" n="167"> +<p> +<s xml:id="echoid-s565" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s565" xml:space="preserve"> +See pages 5 to 7 of the 'Magisteria' (Add MS 6782, f. 110 to f. 112), which contain similar numerical tables. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f217v" o="217v" n="434"/> +<pb file="add_6782_f218" o="218" n="435"/> +<div xml:id="echoid-div168" type="page_commentary" level="2" n="168"> +<p> +<s xml:id="echoid-s567" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s567" xml:space="preserve"> +A draft for page 2 of the 'Magisteria' (Add MS 6782, f. 109). <lb/> +At the end of the page, Harriot suggestes the notation: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mo>,</mo><mi>n</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mo>,</mo><mi>n</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mo>,</mo><mi>n</mi></mstyle></math>, ... inside small boxes, +or better: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>,</mo><mn>7</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>,</mo><mn>6</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>,</mo><mn>5</mn></mstyle></math>, ... inside small boxes, +for what we now write as: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>n</mi><mn>7</mn></msup></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>n</mi><mn>6</mn></msup></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>n</mi><mn>5</mn></msup></mrow></mstyle></math>, .... +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s569" xml:space="preserve"> +Magis placet. <lb/> +<lb/>[<emph style="it">tr: +More pleasing. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f218v" o="218v" n="436"/> +<pb file="add_6782_f219" o="219" n="437"/> +<div xml:id="echoid-div169" type="page_commentary" level="2" n="169"> +<p> +<s xml:id="echoid-s570" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s570" xml:space="preserve"> +The calculations of the previous page (Add MS 6782, f. 218) demonstrated numerically. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f219v" o="219v" n="438"/> +<pb file="add_6782_f220" o="220" n="439"/> +<div xml:id="echoid-div170" type="page_commentary" level="2" n="170"> +<p> +<s xml:id="echoid-s572" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s572" xml:space="preserve"> +Canons for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, see page 22 of the 'Magisteria' (Add MS 6782, f. 129). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head88" xml:space="preserve" xml:lang="lat"> +De Canone. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. De Canone. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +On the canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. On the canon for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +</emph>]<lb/> +</head> +<pb file="add_6782_f220v" o="220v" n="440"/> +<pb file="add_6782_f221" o="221" n="441"/> +<pb file="add_6782_f221v" o="221v" n="442"/> +<pb file="add_6782_f222" o="222" n="443"/> +<pb file="add_6782_f222v" o="222v" n="444"/> +<pb file="add_6782_f223" o="223" n="445"/> +<pb file="add_6782_f223v" o="223v" n="446"/> +<pb file="add_6782_f224" o="224" n="447"/> +<pb file="add_6782_f224v" o="224v" n="448"/> +<pb file="add_6782_f225" o="225" n="449"/> +<pb file="add_6782_f225v" o="225v" n="450"/> +<pb file="add_6782_f226" o="226" n="451"/> +<pb file="add_6782_f226v" o="226v" n="452"/> +<pb file="add_6782_f227" o="227" n="453"/> +<pb file="add_6782_f227v" o="227v" n="454"/> +<pb file="add_6782_f228" o="228" n="455"/> +<div xml:id="echoid-div171" type="page_commentary" level="2" n="171"> +<p> +<s xml:id="echoid-s574" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s574" xml:space="preserve"> +A numerical example of the rule for square root of a sum. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f228v" o="228v" n="456"/> +<pb file="add_6782_f229" o="229" n="457"/> +<div xml:id="echoid-div172" type="page_commentary" level="2" n="172"> +<p> +<s xml:id="echoid-s576" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s576" xml:space="preserve"> +The references on this page are to Viète's +<emph style="it">Supplementum geometriæ</emph> (1593). +</s> +<lb/> +<quote xml:lang="lat"> +Propositio III. <lb/> +Si duae lineae rectae à puncto extra circulum eductae ipsum secent, +pars autem exterior primae fit proportionalis inter partem exteriorem secundae & partem interiorem ejusdem: +erit quoque pars exterior secundae proportionalis inter partem exteriorem primae & partem interiorem ejusdem. +</quote> +<lb/> +<quote> +If two straight lines drawn from a point outside a circle cut it in such a way that +the external part of the first is a proportional between the external and internal parts of the second, +the external part of the second will be a proportional between the external and internal parts of the first. +</quote> +<lb/> +<quote xml:lang="lat"> +Propositio IV. <lb/> +Si duae lineae rectae à puncto extra circulum eductae ipsum secent +quod autem fit sub partibus exterioribus eductarum, aequale fit ei quod fit sub intertioribus: +exteriores partes permutatim sumptae, erunt continue proportionales inter partes interiors. +</quote> +<lb/> +<quote> +If two straight lines drawn from a point outside a circle cut it, +and moreover the product of the external parts is equal to that of the internal parts, +the external parts taken in turn will be continued proportionals between the internal parts. +</quote> +<lb/> +<quote xml:lang="lat"> +Propositio V. <lb/> +Datis duabus lineis rectis, invenire inter easdem duas medias continue, proportionales. +</quote> +<lb/> +<quote> +Given two straight lines, to find two mean proportionals between them. +</quote> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head89" xml:space="preserve" xml:lang="lat"> +Vieta. supl. +pag. 14. b. +<lb/>[<emph style="it">tr: +Viète, Supplementum, page 14v. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s578" xml:space="preserve"> +prop. 5.) +<lb/>[<emph style="it">tr: +Proposition 5. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s579" xml:space="preserve"> +prop 3.) +<lb/>[<emph style="it">tr: +Proposition 3. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s580" xml:space="preserve"> +prop: 4.) +<lb/>[<emph style="it">tr: +Proposition 4. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f229v" o="229v" n="458"/> +<pb file="add_6782_f230" o="230" n="459"/> +<pb file="add_6782_f230v" o="230v" n="460"/> +<pb file="add_6782_f231" o="231" n="461"/> +<div xml:id="echoid-div173" type="page_commentary" level="2" n="173"> +<p> +<s xml:id="echoid-s581" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s581" xml:space="preserve"> +This sheet refers to Stevin's <emph style="it">L'arithmétique ... aussi l'algebre</emph> (1585), page 215, +where there is a section entitled 'De l'addition des racines de multinomies radicaux'. +Stevin gave numerical examples, whereas Harrot has worked in letters. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p> +<s xml:id="echoid-s583" xml:space="preserve"> +vide: Stevin. 215. +<lb/>[<emph style="it">tr: +See Stevin, page 215. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f231v" o="231v" n="462"/> +<pb file="add_6782_f232" o="232" n="463"/> +<div xml:id="echoid-div174" type="page_commentary" level="2" n="174"> +<p> +<s xml:id="echoid-s584" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s584" xml:space="preserve"> +Proofs of the inequalities <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>+</mo><mi>c</mi><mi>c</mi><mo>></mo><mn>2</mn><mi>b</mi><mi>c</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>c</mi><mo>></mo><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f232v" o="232v" n="464"/> +<pb file="add_6782_f233" o="233" n="465"/> +<pb file="add_6782_f233v" o="233v" n="466"/> +<pb file="add_6782_f234" o="234" n="467"/> +<div xml:id="echoid-div175" type="page_commentary" level="2" n="175"> +<p> +<s xml:id="echoid-s586" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s586" xml:space="preserve"> +The table at the top gives formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entry in each column of a table generated +from a constant difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, with every column increasing. <lb/> +Entries in column 0 are constant (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>). <lb/> +Entries in column 1 are denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>1</mn></msup></mrow></mstyle></math>. <lb/> +Entries in column 2 are denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>2</mn></msup></mrow></mstyle></math>. <lb/> +and so on; the small numbers are superscripts, not powers. <lb/> +The lower half of the page lists the possible combinations of increreasing columns (c) and decreasing columns (d), +for up to four columns. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head90" xml:lang="lat"> +Ad progressiones. +<lb/>[<emph style="it">tr: +On progressions +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s588" xml:space="preserve"> +Casus differentiarum progressionum. +<lb/>[<emph style="it">tr: +Cases of progressions of differences. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s589" xml:space="preserve"> +c. designat crescentes progressiones: +<lb/>[<emph style="it">tr: +c. denotes increasing progressions +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s590" xml:space="preserve"> +d. decrescentes. +<lb/>[<emph style="it">tr: +d. decreasing +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s591" xml:space="preserve"> +æquationis secundi omnes <lb/> +hoc casus habentur in <lb/> +alijs chartis. +<lb/>[<emph style="it">tr: +all cases of the second equation here are to be found in other sheets. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f234v" o="234v" n="468"/> +<pb file="add_6782_f235" o="235" n="469"/> +<div xml:id="echoid-div176" type="page_commentary" level="2" n="176"> +<p> +<s xml:id="echoid-s592" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s592" xml:space="preserve"> +Rules for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries in six columns of a table generated from a constant difference, +as on Add MS 6782, f 177–178, but here using the notation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> (for sum) instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>v</mi></mstyle></math>, +to indicate that the entries may be thought of as sums (of lower numbered columns) +rather than differences (of higher numbered columns). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f235v" o="235v" n="470"/> +<pb file="add_6782_f236" o="236" n="471"/> +<div xml:id="echoid-div177" type="page_commentary" level="2" n="177"> +<p> +<s xml:id="echoid-s594" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s594" xml:space="preserve"> +Rules for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries in six columns of a table generated from a constant difference, +as on Add MS 6782, f 177–178, but here using the notation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> (for sum) instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>v</mi></mstyle></math>, +to indicate that the entries may be thought of as sums (of lower numbered columns) +rather than differences (of higher numbered columns). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f236v" o="236v" n="472"/> +<pb file="add_6782_f237" o="237" n="473"/> +<div xml:id="echoid-div178" type="page_commentary" level="2" n="178"> +<p> +<s xml:id="echoid-s596" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s596" xml:space="preserve"> +Calculations similar to those set out on Add MS 6782, f. 148. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head91" xml:space="preserve"> +A. +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s598" xml:space="preserve"> +Ad numeros triangulos et illorum progenies. +<lb/>[<emph style="it">tr: +On triangular numbers and their progeny. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f237v" o="237v" n="474"/> +<pb file="add_6782_f238" o="238" n="475"/> +<pb file="add_6782_f238v" o="238v" n="476"/> +<pb file="add_6782_f239" o="239" n="477"/> +<div xml:id="echoid-div179" type="page_commentary" level="2" n="179"> +<p> +<s xml:id="echoid-s599" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s599" xml:space="preserve"> +At the top of the page is a numerical table in which the leftmost column contains sums of fourth powers +(1 + 16 + 81 + ...). <lb/> +Below that is a smaller numerical table in which the leftmost column contains sums of squares. <lb/> +Further down on the right is a numerical table in which the leftmost column contains sums of cubes. <lb/> +Each table lists successive differences until a final (constant) difference is reached. +</s> +<lb/> +<s xml:id="echoid-s600" xml:space="preserve"> +The reference to Maurolico is to his <emph style="it">Arithmeticorum libri duo</emph> (1575). +Pages 52, 63, and 67 contain tables of several kinds of figurate numbers. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s602" xml:space="preserve"> +Ad summam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi><mi>Z</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +For the sum of squares of squares +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s603" xml:space="preserve"> +Vide Maurolicum <lb/> +in Arithmeticis <lb/> +pag. 52. <lb/> +63. <lb/> +67. +<lb/>[<emph style="it">tr: +See Maurolico, in his Arithmetic, pages 52, 63, 67. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s604" xml:space="preserve"> +Ad summam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +For the sum of cubes. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f239v" o="239v" n="478"/> +<pb file="add_6782_f240" o="240" n="479"/> +<div xml:id="echoid-div180" type="page_commentary" level="2" n="180"> +<p> +<s xml:id="echoid-s605" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s605" xml:space="preserve"> +The top of the page shows the working out of the formula for the sum of square-squares. +</s> +<lb/> +<s xml:id="echoid-s606" xml:space="preserve"> +The proposition quoted from Maurolico is from his +<emph style="it">Arithmeticorum libri duo</emph> (1575), +Proposition 58 (page 25): +</s> +<lb/> +<quote xml:lang="lat"> +Omnis trianguli quadratus, aequalis est aggregato cuborum ab unitate +usque ad cubum triangulo collateralem inclusiue sumptorum. +</quote> +<lb/> +<quote> +The square of every triangular number is equal to the sum of cubes from one, +to the cube of the side of the triangular number, all taken together.) +</quote> +<lb/> +<s xml:id="echoid-s607" xml:space="preserve"> +Maurolico gives as an example <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>2</mn><mn>5</mn><mo>=</mo><mn>1</mn><mo>+</mo><mn>8</mn><mo>+</mo><mn>2</mn><mn>7</mn><mo>+</mo><mn>6</mn><mn>4</mn><mo>+</mo><mn>1</mn><mn>2</mn><mn>5</mn></mstyle></math>. +</s> +<lb/> +<s xml:id="echoid-s608" xml:space="preserve"> +At the bottom of the page are formulae for sums of units, lines, squares, cubes, and square-squares. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head92" xml:lang="lat"> +Ad aggregata Z. C. ZZ. &c. +<lb/>[<emph style="it">tr: +Towards the sums of squares, cubes, square-squares, etc. +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s610" xml:space="preserve"> +least +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s611" xml:space="preserve"> +<reg norm="Maurolico" type="abbr">Maurol</reg>. pag. 25 +<lb/>[<emph style="it">tr: +Maurolico page 25 +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s612" xml:space="preserve"> +omnis triangula quadratus <lb/> +æqualis est aggregato cuborum <lb/> +ab unitate usque ad cubum triangulo <lb/> +collateralem incipio sumptae. +<lb/>[<emph style="it">tr: +the square of every triangluar number is equal to the sum of cubes from one, +to the cube of the corresponding triangular number taken from the beginning. +</emph>]<lb/> +</s> +</p> +<p> +<s xml:id="echoid-s613" xml:space="preserve"> +lines æquall. +</s> +<lb/> +<s xml:id="echoid-s614" xml:space="preserve"> +lines æqually [???] +</s> +<lb/> +<s xml:id="echoid-s615" xml:space="preserve"> +squares æqually [???] <lb/> +in their rootes. &c. +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s616" xml:space="preserve"> +Vel: per reductionem. +<lb/>[<emph style="it">tr: +Or: by reduction. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f240v" o="240v" n="480"/> +<pb file="add_6782_f241" o="241" n="481"/> +<pb file="add_6782_f241v" o="241v" n="482"/> +<pb file="add_6782_f242" o="242" n="483"/> +<pb file="add_6782_f242v" o="242v" n="484"/> +<pb file="add_6782_f243" o="243" n="485"/> +<pb file="add_6782_f243v" o="243v" n="486"/> +<div xml:id="echoid-div181" type="page_commentary" level="2" n="181"> +<p> +<s xml:id="echoid-s617" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s617" xml:space="preserve"> +A generalized table of triangular numbers, +in which each entry is the sum of the entry above it and the entry to the left of it. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f244" o="244" n="487"/> +<pb file="add_6782_f244v" o="244v" n="488"/> +<div xml:id="echoid-div182" type="page_commentary" level="2" n="182"> +<p> +<s xml:id="echoid-s619" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s619" xml:space="preserve"> +A 28-row difference table with constant third difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. +The third, second, and first columns begin with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>, repsectively. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head93" xml:lang="lat"> +Ad Differentias differentiarum. &c. +<lb/>[<emph style="it">tr: +On differences of differences etc. +</emph>]<lb/> +</head> +<pb file="add_6782_f245" o="245" n="489"/> +<pb file="add_6782_f245v" o="245v" n="490"/> +<pb file="add_6782_f246" o="246" n="491"/> +<div xml:id="echoid-div183" type="page_commentary" level="2" n="183"> +<p> +<s xml:id="echoid-s621" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s621" xml:space="preserve"> +The table at the top of the page shows the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>C</mi><mo>+</mo><mn>1</mn><mi>r</mi></mstyle></math> (in modern notation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>+</mo><mi>x</mi></mstyle></math>) +evaluated for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>.</mo></mstyle></math> To the right are columns of successive differences +as far as the constant difference 6. <lb/> +Below that are two further tables, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>C</mi><mo>+</mo><mn>2</mn><mi>r</mi></mstyle></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>+</mo><mi>x</mi></mstyle></math>) and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>C</mi><mo>+</mo><mn>3</mn><mi>r</mi></mstyle></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>+</mo><mi>x</mi></mstyle></math>). +The last one is extrapolated upwards to include values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo>-</mo><mn>2</mn></mstyle></math>. <lb/> +At the bottom of the page are formulae for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th entries columns 0 to 4 of a table generated from +a constant difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> in column 0. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f246v" o="246v" n="492"/> +<pb file="add_6782_f247" o="247" n="493"/> +<div xml:id="echoid-div184" type="page_commentary" level="2" n="184"> +<p> +<s xml:id="echoid-s623" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s623" xml:space="preserve"> +The lower part of this page contains some jottings on binary arithmetic: +addition of 10000 and 10010, multiplication of 101 by 111, and numbers from 1 to 16 in binary +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f247v" o="247v" n="494"/> +<pb file="add_6782_f248" o="248" n="495"/> +<head xml:id="echoid-head94" xml:space="preserve" xml:lang="lat"> +A. Data media trium proportionalium et differentia extremorum invenire extremas. +<lb/>[<emph style="it">tr: +Given the mean of three proportionals and the difference of the extremes, find the extremes. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s625" xml:space="preserve"> +Zet. 2,3: +<lb/>[<emph style="it">tr: +Zetetic II.3 +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s626" xml:space="preserve"> +z. 1,1: +<lb/>[<emph style="it">tr: +Zetetic I.1 +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s627" xml:space="preserve"> +Zet. 2,3: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math> aggregatum extremarum +<lb/>[<emph style="it">tr: +Zetetic II.3: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math> sum of the extrmes +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s628" xml:space="preserve"> +z. 1,1: Tum: +<lb/>[<emph style="it">tr: +Zetetic I.1: then: +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s629" xml:space="preserve"> +Zet. 2,3: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math> aggregatum extremarum +<lb/>[<emph style="it">tr: +Zetetic II.3: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math> sum of the extremes +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s630" xml:space="preserve"> +zet. 1,1. tum: +<lb/>[<emph style="it">tr: +Zetetic I.1, then: +</emph>]<lb/> +</s> +</p> +<head xml:id="echoid-head95" xml:space="preserve" xml:lang="lat"> +A. Data media trium proportionalium et summa extremorum invenire extremas. +<lb/>[<emph style="it">tr: +Given the mean of three proportionals and the sum of the extremes, find the extremes. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s631" xml:space="preserve"> +Tum est <lb/> +supra +<lb/>[<emph style="it">tr: +Then is the above +</emph>]<lb/> +</s> +<s xml:id="echoid-s632" xml:space="preserve"> +Ergo: <lb/> +Zet. 2,4. +<lb/>[<emph style="it">tr: +Therefore, Zetetic II.3. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s633" xml:space="preserve"> +Zet. 1,1. tum: +<lb/>[<emph style="it">tr: +Zetetic I.1, then: +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f248v" o="248v" n="496"/> +<pb file="add_6782_f249" o="249" n="497"/> +<pb file="add_6782_f249v" o="249v" n="498"/> +<div xml:id="echoid-div185" type="page_commentary" level="2" n="185"> +<p> +<s xml:id="echoid-s634" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s634" xml:space="preserve"> +On this page Harriot solves <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mo>=</mo><mn>1</mn><mn>0</mn></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mo>=</mo><mn>8</mn></mstyle></math>. +The latter has real roots 4 and 2, but the roots of the former are complex, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mo>±</mo><msqrt><mrow><mo>-</mo><mn>1</mn></mrow></msqrt></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f250" o="250" n="499"/> +<div xml:id="echoid-div186" type="page_commentary" level="2" n="186"> +<p> +<s xml:id="echoid-s636" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s636" xml:space="preserve"> +At the bottom of this page Harriot solves <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>d</mi><mi>d</mi><mo>=</mo><mn>2</mn><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, +giving the solutions <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>d</mi><mo>±</mo><msqrt><mrow><mo>-</mo><mi>d</mi><mi>d</mi></mrow></msqrt></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f250v" o="250v" n="500"/> +<p> +<s xml:id="echoid-s638" xml:space="preserve"> +Bombell +</s> +</p> +<pb file="add_6782_f251" o="251" n="501"/> +<pb file="add_6782_f251v" o="251v" n="502"/> +<pb file="add_6782_f252" o="252" n="503"/> +<pb file="add_6782_f252v" o="252v" n="504"/> +<pb file="add_6782_f253" o="253" n="505"/> +<pb file="add_6782_f253v" o="253v" n="506"/> +<pb file="add_6782_f254" o="254" n="507"/> +<div xml:id="echoid-div187" type="page_commentary" level="2" n="187"> +<p> +<s xml:id="echoid-s639" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s639" xml:space="preserve"> +The table at the top of the page shows the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>C</mi><mo>+</mo><mn>1</mn><mi>z</mi><mo>+</mo><mn>1</mn><mi>r</mi></mstyle></math> +(in modern notation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>+</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>+</mo><mi>x</mi></mstyle></math>), +evaluated for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn></mstyle></math>. To the right are columns of successive differences +as far as the constant difference 6. <lb/> +Below that are three further tables for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>C</mi><mo>+</mo><mn>2</mn><mi>z</mi><mo>+</mo><mn>3</mn><mi>r</mi></mstyle></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>+</mo><mn>2</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>3</mn><mi>x</mi></mstyle></math>), +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>C</mi><mo>+</mo><mn>3</mn><mi>z</mi><mo>+</mo><mn>2</mn><mi>r</mi></mstyle></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>+</mo><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>2</mn><mi>x</mi></mstyle></math>), and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>C</mi><mo>+</mo><mn>4</mn><mi>z</mi><mo>+</mo><mn>5</mn><mi>r</mi></mstyle></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>+</mo><mn>4</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>5</mn><mi>x</mi></mstyle></math>). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f254v" o="254v" n="508"/> +<pb file="add_6782_f255" o="255" n="509"/> +<pb file="add_6782_f255v" o="255v" n="510"/> +<pb file="add_6782_f256" o="256" n="511"/> +<div xml:id="echoid-div188" type="page_commentary" level="2" n="188"> +<p> +<s xml:id="echoid-s641" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s641" xml:space="preserve"> +The reference on this page is to Proposition 16 from Viète's +<emph style="it">Supplementum geometriæ</emph> (1593). +</s> +<lb/> +<quote xml:lang="lat"> +Proposition XVI. <lb/> +Si duo triangula fuerint aequicrura singula, & ipsa alterum alteri cruribus aequalia, +angulus autem qui est ad basin secundi sit triplus anguli qui est ad basin primi: +cubus ex base primi, minus triplo solido sub base primi & cruris communis quadrato, +aequalis est solido sub base secundi & ejusdem cruris quadrato. +</quote> +<lb/> +<quote> +If two triangles are each isosceles, the legs of one equal to the legs of the other, +and moreover the angle at the base of the second is three times the angle at the base of the first, +then the cube of the first base, minus three times the product of the base of the first and the square of the common side, +is equal to the product of the second base and the square of the same side. +</quote> +<lb/> +<s xml:id="echoid-s642" xml:space="preserve"> +For Harriot's statement of Proposition 16, and a geometric version of the proof, see Add MS 6784, f. 351. +Here he works the proposition algebraically. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head96" xml:space="preserve"> +prop. 16. Supplementi. +<lb/>[<emph style="it">tr: +Proposition 16 from the Supplement +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s644" xml:space="preserve"> +duplicature <lb/> +cubus <lb/>[<emph style="it">tr: +the cube is doubled. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f256v" o="256v" n="512"/> +<pb file="add_6782_f257" o="257" n="513"/> +<pb file="add_6782_f257v" o="257v" n="514"/> +<pb file="add_6782_f258" o="258" n="515"/> +<pb file="add_6782_f258v" o="258v" n="516"/> +<pb file="add_6782_f259" o="259" n="517"/> +<pb file="add_6782_f259v" o="259v" n="518"/> +<pb file="add_6782_f260" o="260" n="519"/> +<pb file="add_6782_f260v" o="260v" n="520"/> +<pb file="add_6782_f261" o="261" n="521"/> +<pb file="add_6782_f261v" o="261v" n="522"/> +<pb file="add_6782_f262" o="262" n="523"/> +<head xml:id="echoid-head97" xml:space="preserve" xml:lang="lat"> +1.) Apotome ex linea secta in extrema et media <lb/> +ratione. +<lb/>[<emph style="it">tr: +An apotome from cutting a line in extreme and mean ratio +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s645" xml:space="preserve"> +Apotome 5<emph style="super">ta</emph>. +<lb/>[<emph style="it">tr: +A fifth apotome. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s646" xml:space="preserve"> +Apotome 1<emph style="super">a</emph>. +<lb/>[<emph style="it">tr: +A first apotome. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s647" xml:space="preserve"> +Apotome 1<emph style="super">a</emph>. +<lb/>[<emph style="it">tr: +A first apotome. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s648" xml:space="preserve"> +Apot: 5<emph style="super">a</emph>. Apot: 1<emph style="super">a</emph>. +<lb/>[<emph style="it">tr: +A fifth apotome. A first apotome. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s649" xml:space="preserve"> +Bin: 5. Apot: 5. +<lb/>[<emph style="it">tr: +A fifth binome. A fifth apotome. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s650" xml:space="preserve"> +Bin: 1<emph style="super">a</emph>. Bin: 5<emph style="super">a</emph>. +<lb/>[<emph style="it">tr: +A first binome. A fifth apotome. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f262v" o="262v" n="524"/> +<pb file="add_6782_f263" o="263" n="525"/> +<div xml:id="echoid-div189" type="page_commentary" level="2" n="189"> +<p> +<s xml:id="echoid-s651" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s651" xml:space="preserve"> +Continued from Add MS 6782, f. 262. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head98" xml:space="preserve" xml:lang="lat"> +2.) De linea secta extrema e media ratione +<lb/>[<emph style="it">tr: +On a line cut in extreme and mean ratio +</emph>]<lb/> +</head> +<pb file="add_6782_f263v" o="263v" n="526"/> +<pb file="add_6782_f264" o="264" n="527"/> +<div xml:id="echoid-div190" type="page_commentary" level="2" n="190"> +<p> +<s xml:id="echoid-s653" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s653" xml:space="preserve"> +This page contains an analysis of Proposition 1 from Book XIII of Euclid's <emph style="it">Elements</emph>: +</s> +<lb/> +<quote> +XIII.1 If a straight line is cut in extreme and mean ratio, +then the square on the greater segment added to the half of the whole is five times the square on the half. +</quote> +<lb/> +<s xml:id="echoid-s654" xml:space="preserve"> +The left hand column gives the 'analysis' or 'resolution' of the problem, +beginning from the final statement and working backwards to discover what conditions must hold. <lb/> +The right hand column gives the 'synthesis' or 'composition', +beginning from the given conditions and working forward to the proof of the proposition. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head99" xml:space="preserve" xml:lang="lat"> +Euclid: lib: 13 +<lb/>[<emph style="it">tr: +Euclid Book XIII +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s656" xml:space="preserve"> +prop. 1. analysis +<lb/>[<emph style="it">tr: +Proposition 1, analysis +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s657" xml:space="preserve"> +sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> secta extra: & med. <lb/> +in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> be cut in extrme and mean ratio in point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s658" xml:space="preserve"> +dico quod +<lb/>[<emph style="it">tr: +I say that +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s659" xml:space="preserve"> +Resolutio +<lb/>[<emph style="it">tr: +Resolution +</emph>]<lb/> +</s> +<lb/> +<lb/>[...]<lb/> +<lb/> +<s xml:id="echoid-s660" xml:space="preserve"> +Est igitur <lb/> +est enim: +<lb/>[<emph style="it">tr: +Therefore it is so; for it is so. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s661" xml:space="preserve"> +Compositio +<lb/>[<emph style="it">tr: +Composition +</emph>]<lb/> +</s> +<lb/> +<lb/>[...]<lb/> +<lb/> +<s xml:id="echoid-s662" xml:space="preserve"> +Quod demonstrare <lb/> +oportuit. +<lb/>[<emph style="it">tr: +Which was to be demonstrated. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f264v" o="264v" n="528"/> +<pb file="add_6782_f265" o="265" n="529"/> +<div xml:id="echoid-div191" type="page_commentary" level="2" n="191"> +<p> +<s xml:id="echoid-s663" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s663" xml:space="preserve"> +The irrationals defined by Euclid in Book X of the <emph style="it">Elements</emph> +are binomes, bimedials, and so on. For their definitions and properties see Add MS 6783, f. 356v to f. 343v. +Here Harriot defines some further irrational quantities, all of them involving fourth roots, +which do not fall into any of Euclid's categories. See also Add MS 6782, f. 266. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head100" xml:space="preserve" xml:lang="lat"> +De speciebus irrationalium ab Euclide omissis +<lb/>[<emph style="it">tr: +On types of irrationals missed by Eculid +</emph>]<lb/> +</head> +<pb file="add_6782_f265v" o="265v" n="530"/> +<pb file="add_6782_f266" o="266" n="531"/> +<div xml:id="echoid-div192" type="page_commentary" level="2" n="192"> +<p> +<s xml:id="echoid-s665" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s665" xml:space="preserve"> +The irrationals defined by Euclid in Book X of the <emph style="it">Elements</emph> +are binomes, bimedials, and so on. For their definitions and properties see Add MS 6783, f. 356v to f. 343v. +Here Harriot defines some further irrational quantities, all of them involving fourth roots, +which do not fall into any of Euclid's categories. See also Add MS 6782, f. 265. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head101" xml:space="preserve" xml:lang="lat"> +De speciebus irrationalium ab Euclide omissis +<lb/>[<emph style="it">tr: +On types of irrationals missed by Eculid +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s667" xml:space="preserve"> +Nota +<lb/>[<emph style="it">tr: +Note +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s668" xml:space="preserve"> +Animadvertendum quod quælibet harum <emph style="st">specier</emph> <emph style="super">irrationalium</emph> +producit quadratum <lb/> +trinomium compositum ex binomio et mediali. +</s> +<s xml:id="echoid-s669" xml:space="preserve"> +Et quodlibet bino-<lb/> +mium huius<emph style="super">modi</emph> speciei <emph style="super">logisticæ</emph> continet in se +implicite duas subspecies. +</s> +<s xml:id="echoid-s670" xml:space="preserve"> +Quod <lb/> +si <emph style="super">in singulis</emph> distincte explica<emph style="super">tur</emph>, +ex istis 5 irrationalibus fient 10. +</s> +<s xml:id="echoid-s671" xml:space="preserve"> +Ut <lb/> +alijs chartis sequentibus apparebit. +<lb/>[<emph style="it">tr: +It is to be noted that any of these irrationals squared produces a trinomial composed of a binome and a medial. +And any binome of this form in letters contains in itself two subforms. +Which if in each case are set out, from these five irrationals there arise 10. +As will appear in the following sheets. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f266v" o="266v" n="532"/> +<pb file="add_6782_f267" o="267" n="533"/> +<div xml:id="echoid-div193" type="page_commentary" level="2" n="193"> +<p> +<s xml:id="echoid-s672" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s672" xml:space="preserve"> +In modern notation, binomes are numbers of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>m</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>n</mi></mrow></msqrt></mstyle></math> +where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> are integers. <lb/> +In Book X, Definitions II, Euclid defined six kinds of binomes, +according to various relationships of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>, +which for Euclid were geometric lengths (see Heath, III, 5–6 and 101–115). +In modern notation, the six binomes may be defined as follows. <lb/> +Binome 1: a binome of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mo>+</mo><msqrt><mrow><mi>n</mi></mrow></msqrt></mstyle></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mo>></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>></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>></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>></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>></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>></mo><msqrt><mrow><mi>n</mi></mrow></msqrt></mstyle></math>, +and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mo>=</mo><mi>n</mi><mo>+</mo><mi>k</mi></mstyle></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><msqrt><mrow><mi>m</mi></mrow></msqrt></mrow><mrow><msqrt><mrow><mi>k</mi></mrow></msqrt></mrow></mfrac></mstyle></math> is non-rational; for example <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>3</mn></mrow></msqrt><mo>+</mo><msqrt><mrow><mn>2</mn></mrow></msqrt></mstyle></math>. <lb/> +Harriot made two further distinctions for binomes of the fifth and sixth kind according to whether +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math> is itself a square (type i) or not (type (ii). <lb/> +In this and the following folio, Add MS 6782, f. 268, +Harriot shows that the square of any binome is always a binome of the first kind. +This folio shows his working for first, second, and third binomes. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head102" xml:space="preserve" xml:lang="lat"> +Binomiorum quadrata, sunt binomia prima. +<lb/>[<emph style="it">tr: +Squares of binomes are binomes of the first kind. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s674" xml:space="preserve"> +1. bin) +<lb/>[<emph style="it">tr: +A binomial of the first kind. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s675" xml:space="preserve"> +2.) +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s676" xml:space="preserve"> +3.) +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s677" xml:space="preserve"> +ut supra. +<lb/>[<emph style="it">tr: +as above. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f267v" o="267v" n="534"/> +<pb file="add_6782_f268" o="268" n="535"/> +<div xml:id="echoid-div194" type="page_commentary" level="2" n="194"> +<p> +<s xml:id="echoid-s678" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s678" xml:space="preserve"> +This folio is the continuation of Add MS 6782, f. 267. +Here Harriot checks that the squares of fourth, fifth, and sixth binomes, +are always binomes of the first kind. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head103" xml:space="preserve" xml:lang="lat"> +Binomiorum quadrata, sunt binomia prima. +<lb/>[<emph style="it">tr: +Squares of binomes are binomes of the first kind. +</emph>]<lb/> +</head> +<pb file="add_6782_f268v" o="268v" n="536"/> +<pb file="add_6782_f269" o="269" n="537"/> +<pb file="add_6782_f269v" o="269v" n="538"/> +<pb file="add_6782_f270" o="270" n="539"/> +<pb file="add_6782_f270v" o="270v" n="540"/> +<pb file="add_6782_f271" o="271" n="541"/> +<pb file="add_6782_f271v" o="271v" n="542"/> +<pb file="add_6782_f272" o="272" n="543"/> +<p xml:lang="lat"> +<s xml:id="echoid-s680" xml:space="preserve"> +Data secundum trium proportionalium: invenire primam et tertiam, <lb/> +ut illarum differentia sit æqualis bis secundæ datæ. +<lb/>[<emph style="it">tr: +Given the second of three proportionals: +find the first and third so that their difference is equal to twice the given second. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s681" xml:space="preserve"> +Sit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> prima, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, secunda. <lb/> +ut illarum differentia sit æqualis bis secundæ datæ. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the first, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> the second, such that their difference is equal to twice the given second. +</emph>]<lb/> +</s> +<lb/> +<lb/>[...]<lb/> +<lb/> +<s xml:id="echoid-s682" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>a</mi><mi>t</mi><mo>=</mo><mi>a</mi><mi>c</mi></mstyle></math>. Rationalis potentia. +<lb/>[<emph style="it">tr: +Rational in square. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s683" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> Apotome 5<emph style="super">ta</emph>, 1<emph style="super">o</emph> +<lb/>[<emph style="it">tr: +A fifth apotome +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s684" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math> Rationalis posita. +<lb/>[<emph style="it">tr: +The supposed rational +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s685" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>f</mi></mstyle></math> Binomia 5<emph style="super">a</emph>, 1<emph style="super">o</emph> +<lb/>[<emph style="it">tr: +A fifth binome +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s686" xml:space="preserve"> +Erit etiam <lb/>[...]<lb/> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math>. Binomia 4<emph style="super">a</emph>. +<lb/>[<emph style="it">tr: +There will also be <lb/>[...]<lb/> a fourth binome +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s687" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>b</mi></mstyle></math>. Apotome 4<emph style="super">a</emph>. +<lb/>[<emph style="it">tr: +a fourth apotome +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s688" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>e</mi></mstyle></math>. cum rationalium medium totum efficiens 1<emph style="super">o</emph> +<lb/>[<emph style="it">tr: +with the rational, making the mean of all +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s689" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi></mstyle></math>. Minor. +<lb/>[<emph style="it">tr: +Lesser +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s690" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>e</mi></mstyle></math>. Media. +<lb/>[<emph style="it">tr: +Mean +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s691" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math>. Rationale et medium potens. 1<emph style="super">o</emph>. +<lb/>[<emph style="it">tr: +A power of the rational and the mean +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f272v" o="272v" n="544"/> +<pb file="add_6782_f273" o="273" n="545"/> +<pb file="add_6782_f273v-274" o="273v-274" n="546"/> +<pb file="add_6782_f273v" o="273v" n="547"/> +<pb file="add_6782_f274" o="274" n="548"/> +<pb file="add_6782_f274v" o="274v" n="549"/> +<pb file="add_6782_f275" o="275" n="550"/> +<pb file="add_6782_f275v" o="275v" n="551"/> +<pb file="add_6782_f276" o="276" n="552"/> +<div xml:id="echoid-div195" type="page_commentary" level="2" n="195"> +<p> +<s xml:id="echoid-s692" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s692" xml:space="preserve"> +Powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo maxsize="1">)</mo></mstyle></math> up to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mrow><msup><mo maxsize="1">)</mo><mn>5</mn></msup></mrow></mstyle></math>. +Each power is calculated from the previous one by multiplication. <lb/> +Note the use of cossist notation: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> for a first power, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math> for a square, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> for a cube, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>z</mi></mstyle></math> for a square-suare or fourth power, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>ß</mo></mstyle></math> for a sursolid or fifth power. <lb/> +Below the main table is a list of the final sums, including the sixth power (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>c</mi></mstyle></math>), +which has not been calculated on this page +but which can be deduced from the pattern for the previous cases. <lb/> +For a similar table see Add MS 6786, f. 457. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s694" xml:space="preserve"> +Forma <emph style="st">generationis continue</emph> +<emph style="super">generandi figurata</emph> <lb/> +<emph style="st">proportionalium ab unitate</emph> +<emph style="super">in binomia radice</emph> <lb/> +per logisticen speciosam: <lb/> +<emph style="st">ad demonstrandum pro parte alium</emph> <lb/> +<emph style="st">in numeris analysin.</emph> +<lb/>[<emph style="it">tr: +A method of generating figurate numbers from binomial roots in letters: +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s695" xml:space="preserve"> +Nota: pro porismo. +<lb/>[<emph style="it">tr: +Note: for the proof. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s696" xml:space="preserve"> +Species partium <lb/> +unius cuisque potentiæ <lb/> +sunt continue proportionales <lb/> +ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +The case of a single part where the powers are in continued proportion as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +</emph>]<lb/> +</s> +<s xml:id="echoid-s697" xml:space="preserve"> +Et in numeris, sunt <lb/> +termini minimi si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> sunt primi &c. <lb/> +et non in <lb/> +ratione <lb/> +multiplicant. +<lb/>[<emph style="it">tr: +And in numbers, these are the lowest terms, if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> are the first, <lb/> +and they are not multiplied by some ratio. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f276v" o="276v" n="553"/> +<pb file="add_6782_f277" o="277" n="554"/> +<div xml:id="echoid-div196" type="page_commentary" level="2" n="196"> +<p> +<s xml:id="echoid-s698" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s698" xml:space="preserve"> +This page shows notation used for first, second, third, ...., ninth powers in the following authors: <lb/> +Diophantus in <emph style="it">Diophanti Alexandrini rerum arithmeticarum libri sex</emph>, +edited by Wilhelm Xylander (1575); <lb/> +François Viète in, for example, <emph style="it">In artem analyticen isagoge</emph> (1591); <lb/> +Bernard Salignac in <emph style="it">Arithmeticae libri duo et algebrae totidem</emph> (1580, 1593); <lb/> +Michael Stifel in <emph style="it">Arithmetica integra</emph> (1544); <lb/> +Christoph Clavius in <emph style="it">Algebra</emph> (1608); <lb/> +Simon Stevin in <emph style="it">L'arithmétique ... aussi l'algèbre</emph> (1585). <lb/> +The inclusion of Clavius in this list is particularly significant since it dates the page to 1608 or later. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s700" xml:space="preserve"> +& indices gradarum +<t> +ec. indices of the degrees +</t> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s701" xml:space="preserve"> +ut Diophantus et Vieta +<t> +as in Diophantus and Viète +</t> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s702" xml:space="preserve"> +ut Salignacus +<t> +as in Salignacus +</t> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s703" xml:space="preserve"> +ut Stifelius, Clavius et alij +<t> +as in Stifel, Clavius and others +</t> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s704" xml:space="preserve"> +ut Stevinus et alij +<t> +as in Stevin and others +</t> +</s> +</p> +<pb file="add_6782_f277v" o="277v" n="555"/> +<pb file="add_6782_f278" o="278" n="556"/> +<div xml:id="echoid-div197" type="page_commentary" level="2" n="197"> +<p> +<s xml:id="echoid-s705" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s705" xml:space="preserve"> +Powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mn>6</mn><mn>0</mn><mo>+</mo><mn>7</mn><mo maxsize="1">)</mo></mstyle></math> up to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mn>6</mn><mn>0</mn><mo>+</mo><mn>7</mn><mrow><msup><mo maxsize="1">)</mo><mn>5</mn></msup></mrow></mstyle></math> following the pattern laid out in f. 276. <lb/> +A calculation below each box gives the sum of the figures contained in it. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f278v" o="278v" n="557"/> +<pb file="add_6782_f279" o="279" n="558"/> +<div xml:id="echoid-div198" type="page_commentary" level="2" n="198"> +<p> +<s xml:id="echoid-s707" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s707" xml:space="preserve"> +The example <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mn>6</mn><mn>0</mn><mo>+</mo><mn>7</mn><mrow><msup><mo maxsize="1">)</mo><mn>3</mn></msup></mrow></mstyle></math> from f. 278 set out to show how the binomial coefficients are used. +Thus, in calculating the cube, for wich the coefficients are 1,3, 3, 1, +the cube of 6 (the relevant digit of 60) is used once, +the square is taken 3 times and multiplied by 7, +while 6 is also taken 3 times and multiplied by the square of 7; +finally the cube of 7 is added once. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s709" xml:space="preserve"> +Numerorum dispositio <lb/> +ad figuratorum genesin et <lb/> +analysin demonstrandam. +<lb/>[<emph style="it">tr: +The disposition of the numbers for the generation of figurate numbers and for demonstrating the analysis. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f279v" o="279v" n="559"/> +<pb file="add_6782_f280" o="280" n="560"/> +<div xml:id="echoid-div199" type="page_commentary" level="2" n="199"> +<p> +<s xml:id="echoid-s710" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s710" xml:space="preserve"> +The upper third of the page contains calculations of powers of 24, up to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mn>2</mn><mn>4</mn><mrow><msup><mo maxsize="1">)</mo><mn>5</mn></msup></mrow></mstyle></math>. <lb/> +The lower third of the page contain calculations of powers of 67 (see also f. 278). <lb/> +In the middle of the page, the binomial coefficients are listed in three different layouts. +The table on the left shows how each row may be calculated by adding two copies of the previous row. +A similar table appears again in the lower right of the page. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f280v" o="280v" n="561"/> +<div xml:id="echoid-div200" type="page_commentary" level="2" n="200"> +<p> +<s xml:id="echoid-s712" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s712" xml:space="preserve"> +The first units mentioned are bushels, a measure of grain, equivalent to 4 pecks or 8 gallons. <lb/> +The page contains a conversion of 6553600000 bushels per square mile to 10485760 bushels per acre +(1 square mile = 640 acres), +and a conversion of 262144 acres to 409 square miles. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p> +<s xml:id="echoid-s714" xml:space="preserve"> +10485,760 bushelles +</s> +</p> +<p> +<s xml:id="echoid-s715" xml:space="preserve"> +262,144 acres +</s> +</p> +<p> +<s xml:id="echoid-s716" xml:space="preserve"> +640 acres in <lb/> +a square mile +</s> +</p> +<p> +<s xml:id="echoid-s717" xml:space="preserve"> +409 miles square +</s> +</p> +<p> +<s xml:id="echoid-s718" xml:space="preserve"> +20 miles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>4</mn></mrow><mrow><mn>5</mn></mrow></mfrac></mstyle></math> +</s> +</p> +<pb file="add_6782_f281" o="281" n="562"/> +<div xml:id="echoid-div201" type="page_commentary" level="2" n="201"> +<p> +<s xml:id="echoid-s719" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s719" xml:space="preserve"> +Here Harriot calculates the square root of 4489, the cube root of 300763, +the fourth root of 20151121, and the sixth root of 1350125107, +demonstrating that the answer is 67 in each case. +This is the analysis, or taking apart, of what has been constructed on f. 278. <lb/> +Maurolico's treatment of cube roots begins on page 110 of his +<emph style="it">Arithmeticorum libri duo</emph> (1575). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head104" xml:lang="lat"> +Analysis: +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s721" xml:space="preserve"> +Inde: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi><mi>d</mi><mo>=</mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mi>d</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math> <lb/> +ut Maurolicus et nos in alia charta demonstravimus. +<lb/>[<emph style="it">tr: +Whence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi><mi>d</mi><mo>=</mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mi>d</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>, as in Maurolicus and as I have demonstrated in another sheet. +</emph>]<lb/> +<sc> +This note shows an alternative method of calculation, attributed to Maurolico, +in which <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> is replaced by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. +An asterisk against the note directs the reader to Maurolico's method of calculation, on the right. +</sc> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s722" xml:space="preserve"> +Maurolicus <lb/> +[???] +</s> +</p> +<pb file="add_6782_f281v" o="281v" n="563"/> +<pb file="add_6782_f282" o="282" n="564"/> +<pb file="add_6782_f282v" o="282v" n="565"/> +<pb file="add_6782_f283" o="283" n="566"/> +<pb file="add_6782_f283v" o="283v" n="567"/> +<pb file="add_6782_f284" o="284" n="568"/> +<pb file="add_6782_f284v" o="284v" n="569"/> +<pb file="add_6782_f285" o="285" n="570"/> +<pb file="add_6782_f285v" o="285v" n="571"/> +<p xml:lang="lat"> +<s xml:id="echoid-s723" xml:space="preserve"> +oppose?? <lb/> +suum coniugatum <lb/> +eosdem habet <lb/> +numeros. +<lb/>[<emph style="it">tr: +their conjugates have the same numbers. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s724" xml:space="preserve"> +contrarium est <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mo>-</mo><mo>=</mo><mo>+</mo></mstyle></math>. +<lb/>[<emph style="it">tr: +the opposite is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mo>-</mo><mo>=</mo><mo>+</mo></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s725" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>r</mi></mstyle></math> quando: +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>r</mi></mstyle></math> when: +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s726" xml:space="preserve"> +ut in <lb/> +2<emph style="super">o</emph> et 4<emph style="super">o</emph> casu <lb/> +et ut in alia <lb/> +charta probatur <lb/> +universaliter. +<lb/>[<emph style="it">tr: +as in the 2nd and 4th cases and as it is proved generally in the other sheet. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s727" xml:space="preserve"> +tum <emph style="st">habetur</emph> unum (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>) ex consequenti <lb/> +erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mn>2</mn><mi>r</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +then as a consequence one value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>r</mi></mstyle></math>. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s728" xml:space="preserve"> +alterum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> habetur ex universalis <lb/> +methodo in alia charta. +<lb/>[<emph style="it">tr: +another value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> is to be had by the general method in another sheet. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f286" o="286" n="572"/> +<pb file="add_6782_f286v" o="286v" n="573"/> +<pb file="add_6782_f287" o="287" n="574"/> +<pb file="add_6782_f287v" o="287v" n="575"/> +<pb file="add_6782_f288" o="288" n="576"/> +<pb file="add_6782_f288v" o="288v" n="577"/> +<pb file="add_6782_f289" o="289" n="578"/> +<pb file="add_6782_f289v" o="289v" n="579"/> +<pb file="add_6782_f290" o="290" n="580"/> +<pb file="add_6782_f290v" o="290v" n="581"/> +<pb file="add_6782_f291" o="291" n="582"/> +<pb file="add_6782_f291v" o="291v" n="583"/> +<pb file="add_6782_f292" o="292" n="584"/> +<pb file="add_6782_f292v" o="292v" n="585"/> +<pb file="add_6782_f293" o="293" n="586"/> +<pb file="add_6782_f293v" o="293v" n="587"/> +<pb file="add_6782_f294" o="294" n="588"/> +<pb file="add_6782_f294v" o="294v" n="589"/> +<pb file="add_6782_f295" o="295" n="590"/> +<pb file="add_6782_f295v" o="295v" n="591"/> +<pb file="add_6782_f296" o="296" n="592"/> +<pb file="add_6782_f296v" o="296v" n="593"/> +<pb file="add_6782_f297" o="297" n="594"/> +<pb file="add_6782_f297v" o="297v" n="595"/> +<pb file="add_6782_f298" o="298" n="596"/> +<div xml:id="echoid-div202" type="page_commentary" level="2" n="202"> +<p> +<s xml:id="echoid-s729" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s729" xml:space="preserve"> +On this folio Harriot gives rules for finding all the parameters of an arithmetic progression +given any three of them. <lb/> +The three parameters supposed given are listed in the second column, headed 'data', +where Harriot runs systematically through all the combinations <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>u</mi><mi>n</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>u</mi><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>u</mi><mi>s</mi></mstyle></math>, and so on. <lb/> +Rules for finding the two remaining quantities in each case are given in the third column, headed 'quaesita'. <lb/> +For further details see Stedall 2007. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head105" xml:lang="lat"> +Omnes casus arithmeticæ progressionis simplicis <lb/> +primi ordinis +<lb/>[<emph style="it">tr: +All cases of simple arithmetic progressions of the first order +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s731" xml:space="preserve"> +casus. data. quæsita. +<lb/>[<emph style="it">tr: +case. given. sought. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s732" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. primus numeris. +<lb/>[<emph style="it">tr: +first number. +</emph>]<lb/> +</s> +<s xml:id="echoid-s733" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. ultimus. +<lb/>[<emph style="it">tr: +last number. +</emph>]<lb/> +</s> +<s xml:id="echoid-s734" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. differentia. <lb/> +<lb/>[<emph style="it">tr: +difference. +</emph>]<lb/> +</s> +<s xml:id="echoid-s735" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>. numeris locorum. +<lb/>[<emph style="it">tr: +number of places. +</emph>]<lb/> +</s> +<s xml:id="echoid-s736" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>. summa. +<lb/>[<emph style="it">tr: +sum. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s737" xml:space="preserve"> +6 et 9 casus qui <lb/> +signantur * <lb/> +soluuntur æquationibus <lb/> +quadraticis +<lb/>[<emph style="it">tr: +Cases 6 and 9 marked thus * are solved by quadratic equations. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s738" xml:space="preserve"> +Ex his terminis <lb/> +tribus datis, dantur reliqui. +<lb/>[<emph style="it">tr: +From any three terms given, the rest may be found. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s739" xml:space="preserve"> +I. unitas. +<lb/>[<emph style="it">tr: +I. a unit. +</emph>]<lb/> +</s> +<s xml:id="echoid-s740" xml:space="preserve"> +II. quadratum unitatis. +<lb/>[<emph style="it">tr: +II. the square of a unit. +</emph>]<lb/> +</s> +<s xml:id="echoid-s741" xml:space="preserve"> +III. cubus unitatis. +<lb/>[<emph style="it">tr: +III. the cube of a unit. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f298v" o="298v" n="597"/> +<pb file="add_6782_f299" o="299" n="598"/> +<div xml:id="echoid-div203" type="page_commentary" level="2" n="203"> +<p> +<s xml:id="echoid-s742" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s742" xml:space="preserve"> +Rough work, and some equations in words for arithmetic progressions. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head106" xml:lang="lat"> +De progressione Arithmetica. +<lb/>[<emph style="it">tr: +On arithmetic progressions +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s744" xml:space="preserve"> +In Arithmetica progressionis <lb/> +<lb/>[<emph style="it">tr: +In arithmetic progressions +</emph>]<lb/> +</s> +<s xml:id="echoid-s745" xml:space="preserve"> +[1.]) Numerus terminorum – 1 = Numerus differentiorum. <lb/> +<lb/>[<emph style="it">tr: +The number of terms – 1 = the number of differences. +</emph>]<lb/> +</s> +<s xml:id="echoid-s746" xml:space="preserve"> +[2.]) Maximus terminorum – minimo = Summa differentiorum. <lb/> +<lb/>[<emph style="it">tr: +The greatest term – the least term = the sum of the differences. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s747" xml:space="preserve"> +[etc.] +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s748" xml:space="preserve"> +Terminus primorum vel ultimorum +<lb/>[<emph style="it">tr: +The term of the first or the last +</emph>]<lb/> +</s> +<s xml:id="echoid-s749" xml:space="preserve"> +excessus. <lb/> +<lb/>[<emph style="it">tr: +The excess. +</emph>]<lb/> +</s> +<s xml:id="echoid-s750" xml:space="preserve"> +Summa. <lb/> +<lb/>[<emph style="it">tr: +The sum. +</emph>]<lb/> +</s> +<s xml:id="echoid-s751" xml:space="preserve"> +Numerus locorum. <lb/> +<lb/>[<emph style="it">tr: +The number of places. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f299v" o="299v" n="599"/> +<pb file="add_6782_f300" o="300" n="600"/> +<p> +<s xml:id="echoid-s752" xml:space="preserve"> +The ground player <lb/> +The persepctive player +</s> +</p> +<pb file="add_6782_f300v" o="300v" n="601"/> +<pb file="add_6782_f301" o="301" n="602"/> +<pb file="add_6782_f301v" o="301v" n="603"/> +<pb file="add_6782_f302" o="302" n="604"/> +<pb file="add_6782_f302v" o="302v" n="605"/> +<pb file="add_6782_f303" o="303" n="606"/> +<pb file="add_6782_f303v" o="303v" n="607"/> +<pb file="add_6782_f304" o="304" n="608"/> +<pb file="add_6782_f304v" o="304v" n="609"/> +<pb file="add_6782_f305" o="305" n="610"/> +<pb file="add_6782_f305v" o="305v" n="611"/> +<pb file="add_6782_f306" o="306" n="612"/> +<pb file="add_6782_f306v" o="306v" n="613"/> +<pb file="add_6782_f307" o="307" n="614"/> +<pb file="add_6782_f307v" o="307v" n="615"/> +<pb file="add_6782_f308" o="308" n="616"/> +<pb file="add_6782_f308v" o="308v" n="617"/> +<pb file="add_6782_f309" o="309" n="618"/> +<pb file="add_6782_f309v" o="309v" n="619"/> +<pb file="add_6782_f310" o="310" n="620"/> +<pb file="add_6782_f310v" o="310v" n="621"/> +<pb file="add_6782_f311" o="311" n="622"/> +<div xml:id="echoid-div204" type="page_commentary" level="2" n="204"> +<p> +<s xml:id="echoid-s753" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s753" xml:space="preserve"> +This page refers to Stevin's <emph style="it">L'arithmétique … aussi l'algebre</emph> (1585), page 331, +where Stevin discusses the equation 1(3) = 6(2) + 400 (in modern notation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>=</mo><mn>6</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>=</mo><mn>4</mn><mn>0</mn><mn>0</mn></mstyle></math>.) +Here Harriot works on the same equation, written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>3</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>4</mn><mn>0</mn><mn>0</mn></mstyle></math>. See also Add MS 6782, f. 311v. <lb/> +The letters S,WL that appear in this page presumably refer to Harriot's friend Sir William Lower. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p> +<s xml:id="echoid-s755" xml:space="preserve"> +a) Stevin. 331 +</s> +</p> +<p> +<s xml:id="echoid-s756" xml:space="preserve"> +S,WL +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s757" xml:space="preserve"> +Ergo species non est universalis. +<lb/>[<emph style="it">tr: +Therefore the rule is not general. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f311v" o="311v" n="623"/> +<div xml:id="echoid-div205" type="page_commentary" level="2" n="205"> +<p> +<s xml:id="echoid-s758" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s758" xml:space="preserve"> +Further work relating to Add MS 6782, f. 311. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p> +<s xml:id="echoid-s760" xml:space="preserve"> +Stevin. 331 +</s> +</p> +<pb file="add_6782_f312" o="312" n="624"/> +<div xml:id="echoid-div206" type="page_commentary" level="2" n="206"> +<p> +<s xml:id="echoid-s761" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s761" xml:space="preserve"> +Further work based on Stevin's <emph style="it">L 19arithmétique … aussi l 19algebre</emph> (1585), page 331. +Here Harriot works on general equations of the type <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>f</mi><mi>f</mi><mi>f</mi></mstyle></math>. <lb/> +The letters S,WL that appear on this page presumably refer to Harriot's friend Sir William Lower. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p> +<s xml:id="echoid-s763" xml:space="preserve"> +b) Stevin. 331 +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s764" xml:space="preserve"> +species non universalis S,WL +<lb/>[<emph style="it">tr: +The rule is not universal. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f312v" o="312v" n="625"/> +<pb file="add_6782_f313" o="313" n="626"/> +<div xml:id="echoid-div207" type="page_commentary" level="2" n="207"> +<p> +<s xml:id="echoid-s765" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s765" xml:space="preserve"> +The reference on this page is to Aulus Gellius, <emph style="it">Noctes atticae</emph>, (first printed 1469). +Chapter 22 of Book II is entitled +'De vento iapyge deque aliorum ventorum vocabulis regionibusque accepta ex Favorini sermonibus'. +There Aulus Gellius names the winds from each direction; Harriot has placed them around the points of a compass. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head107" xml:space="preserve" xml:lang="lat"> +Ex Aulo Gellio. lib. 2. cap. 22. pag. 63. +</head> +<pb file="add_6782_f313v" o="313v" n="627"/> +<pb file="add_6782_f314" o="314" n="628"/> +<div xml:id="echoid-div208" type="page_commentary" level="2" n="208"> +<p> +<s xml:id="echoid-s767" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s767" xml:space="preserve"> +Canonical forms for equation with three or four positive roots. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head108" xml:space="preserve"> +B) +</head> +<pb file="add_6782_f314v" o="314v" n="629"/> +<pb file="add_6782_f315" o="315" n="630"/> +<div xml:id="echoid-div209" type="page_commentary" level="2" n="209"> +<p> +<s xml:id="echoid-s769" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s769" xml:space="preserve"> +An examination of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>6</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>1</mn><mi>a</mi><mo>=</mo><mn>6</mn></mstyle></math>, which has roots 1, 2, 3. +This is one of several equations with multiple roots treated by Viète in +<emph style="it">De numerosa potestatum resolutione</emph>. +Harriot solves it in full on Add MS 6783, f. 187, and refers to it again in Add MS 6783, f. 188. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head109" xml:space="preserve"> +C) +</head> +<pb file="add_6782_f315v" o="315v" n="631"/> +<div xml:id="echoid-div210" type="page_commentary" level="2" n="210"> +<p> +<s xml:id="echoid-s771" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s771" xml:space="preserve"> +An examination of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>1</mn><mn>2</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>2</mn><mn>9</mn><mi>a</mi><mo>=</mo><mn>1</mn><mn>8</mn></mstyle></math>, which has roots 1, 2, 9. +This is one of several equations with multiple roots treated by Viète in +<emph style="it">De numerosa potestatum resolutione</emph>. +Harriot solves it in full on Add MS 6783, f. 187. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head110" xml:space="preserve"> +D) +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s773" xml:space="preserve"> +Triens coefficientis longituidnis. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math> +<lb/>[<emph style="it">tr: +A third of the longitudinal coefficient +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s774" xml:space="preserve"> +Triplum quadratum. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>b</mi><mi>b</mi><mo>+</mo><mn>6</mn><mi>b</mi><mi>d</mi><mo>+</mo><mn>3</mn><mi>d</mi><mi>d</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Three times the square +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s775" xml:space="preserve"> +maius est coefficientibus planis <lb/> +per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>d</mi><mi>d</mi><mo>+</mo><mn>9</mn><mi>d</mi><mi>c</mi><mo>+</mo><mn>9</mn><mi>c</mi><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +greater than the plane coefficient by +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s776" xml:space="preserve"> +Duplus cubus e triente. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Twice the cube of a third of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math> +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s777" xml:space="preserve"> +maior +<lb/>[<emph style="it">tr: +greater than +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s778" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math> <lb/> +in coefficientibus planibus +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math> times the plane coefficient +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s779" xml:space="preserve"> +Excessus maximi laterus supra <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math> fit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> +<lb/>[<emph style="it">tr: +The excess of the greatest side over <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s780" xml:space="preserve"> +reliqua duo sunt minora <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +the two remaining are less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math> +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s781" xml:space="preserve"> +maxium latus <lb/> +erit +<lb/>[<emph style="it">tr: +the greatest side will be +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s782" xml:space="preserve"> +Examinatio. Vide Charta B) +<lb/>[<emph style="it">tr: +Examination. See sheet B. +</emph>]<lb/> +[<emph style="it">Note: +Sheet B is Add MS 6782, f. 314. + </emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s783" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> in <lb/> +coefficientia <lb/> +plana. +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> times the plane coefficient +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s784" xml:space="preserve"> +Differentia. +<lb/>[<emph style="it">tr: +Difference. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s785" xml:space="preserve"> +Nota. Vide K in D.1. +<lb/>[<emph style="it">tr: +Note. See K in D.1. +</emph>]<lb/> +[<emph style="it">Note: +Sheet D.1 is Add MS 6782, f. 316. + </emph>]<lb/> +</s> +</p> +<pb file="add_6782_f316" o="316" n="632"/> +<head xml:id="echoid-head111" xml:space="preserve"> +D.1. +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s786" xml:space="preserve"> +D Nota K. +<lb/>[<emph style="it">tr: +Note K for sheet D. +</emph>]<lb/> +[<emph style="it">Note: +Sheet D is Add MS 6783, f. 315v. + </emph>]<lb/> +</s> +</p> +<pb file="add_6782_f316v" o="316v" n="633"/> +<pb file="add_6782_f317" o="317" n="634"/> +<div xml:id="echoid-div211" type="page_commentary" level="2" n="211"> +<p> +<s xml:id="echoid-s787" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s787" xml:space="preserve"> +An examination of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>9</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>2</mn><mn>4</mn><mi>a</mi><mo>=</mo><mn>2</mn><mn>0</mn></mstyle></math>, which has roots 2, 2, 5. +This is one of several equations with multiple roots treated by Viète in +<emph style="it">De numerosa potestatum resolutione</emph>. +Harriot solves it in full on Add MS 6783, f. 187. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head112" xml:space="preserve"> +E) +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s789" xml:space="preserve"> +Triens coeff: long: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> +<lb/>[<emph style="it">tr: +A third of the longitudinal coefficient +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s790" xml:space="preserve"> +Triplum quadrat: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>b</mi><mi>b</mi><mo>+</mo><mn>6</mn><mi>b</mi><mi>c</mi><mo>+</mo><mn>3</mn><mi>c</mi><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Three times the square +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s791" xml:space="preserve"> +maius est coefficientibus planis <lb/> +per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>c</mi><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +greater than the plane coefficient by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>c</mi><mi>c</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s792" xml:space="preserve"> +Duplus cubus e triente. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Twice the cube of the third +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s793" xml:space="preserve"> +maior +</s> +<lb/>[<emph style="it">tr: +greater than +</emph>]<lb/> +<lb/> +<s xml:id="echoid-s794" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> in <lb/> +coefficientibus planibus +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> times the plane coefficient +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s795" xml:space="preserve"> +Excessus maximi laterus supra <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> +<lb/>[<emph style="it">tr: +Let the excess of the greatest side over <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s796" xml:space="preserve"> +reliqua duo sunt minora <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +the remaining two are less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math> +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s797" xml:space="preserve"> +maxium latus <lb/> +erit +<lb/>[<emph style="it">tr: +the greatest side will be +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s798" xml:space="preserve"> +Examinatio. +<lb/>[<emph style="it">tr: +Examination +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s799" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> in <lb/> +coeff +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> times the coefficient +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s800" xml:space="preserve"> +Differentia. +<lb/>[<emph style="it">tr: +Difference +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s801" xml:space="preserve"> +Aliter casus +<lb/>[<emph style="it">tr: +Another case +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f317v" o="317v" n="635"/> +<pb file="add_6782_f318" o="318" n="636"/> +<div xml:id="echoid-div212" type="page_commentary" level="2" n="212"> +<p> +<s xml:id="echoid-s802" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s802" xml:space="preserve"> +An examination of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>1</mn><mn>8</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>9</mn><mn>5</mn><mi>a</mi><mo>=</mo><mn>1</mn><mn>2</mn><mn>6</mn></mstyle></math>, which has roots 2, 7, 9. +This is one of several equations with multiple roots treated by Viète in +<emph style="it">De potestatum numerosa resolutione</emph>. +Harriot solves it in full on Add MS 6783, f. 187. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head113" xml:space="preserve"> +F) +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s804" xml:space="preserve"> +Triens coefficientis longituidnis. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>d</mi></mstyle></math> +<lb/>[<emph style="it">tr: +A third of the longitudinal coefficient +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s805" xml:space="preserve"> +Triplum quadratum. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>b</mi><mi>b</mi><mo>-</mo><mn>6</mn><mi>b</mi><mi>d</mi><mo>+</mo><mn>3</mn><mi>d</mi><mi>d</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Three times the square +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s806" xml:space="preserve"> +maius est coefficientibus planis <lb/> +per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>d</mi><mi>d</mi><mo>+</mo><mn>9</mn><mi>c</mi><mi>c</mi><mo>-</mo><mn>9</mn><mi>c</mi><mi>d</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +greater than the plane coefficient by +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s807" xml:space="preserve"> +Duplus cubus e triente. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>d</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Twice the cube of the third +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s808" xml:space="preserve"> +minor +<lb/>[<emph style="it">tr: +less than +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s809" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>d</mi></mstyle></math> <lb/> +in coefficientibus planibus +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>d</mi></mstyle></math> times the plane coefficient +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s810" xml:space="preserve"> +Excessus +<lb/>[<emph style="it">tr: +Excess +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s811" xml:space="preserve"> +medium et maximum latus <lb/> +excedunt. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>d</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +medium and maximum sides exceed +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s812" xml:space="preserve"> +sit unus vel alter excessu, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +let one or other excess be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s813" xml:space="preserve"> +erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mi>d</mi></mstyle></math>, excessus medij 1. adde <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>d</mi></mstyle></math> erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> medium. <lb/> +erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mn>3</mn><mi>c</mi><mo>-</mo><mn>2</mn><mi>d</mi></mstyle></math>, excessus maximi 3. adde <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>d</mi></mstyle></math> erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mn>3</mn><mi>c</mi><mo>-</mo><mn>2</mn><mi>d</mi></mstyle></math> maximum. +<lb/>[<emph style="it">tr: +if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mi>d</mi></mstyle></math>, the excess of the medium, add <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>d</mi></mstyle></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> will be the medium; <lb/> +if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mn>3</mn><mi>c</mi><mo>-</mo><mn>2</mn><mi>d</mi></mstyle></math>, the excess of the maximum, add <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>d</mi></mstyle></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mn>3</mn><mi>c</mi><mo>-</mo><mn>3</mn><mi>d</mi></mstyle></math> will be the maximum; +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s814" xml:space="preserve"> +Examinatio +<lb/>[<emph style="it">tr: +Examination +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s815" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>c</mi></mstyle></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>d</mi></mstyle></math> <lb/> +in coeff planis +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>c</mi></mstyle></math> times <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>d</mi></mstyle></math> times the plane coefficient +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s816" xml:space="preserve"> +cubus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>c</mi><mo>-</mo><mn>2</mn><mi>d</mi></mstyle></math> +<lb/>[<emph style="it">tr: +The cube of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>c</mi><mo>-</mo><mn>2</mn><mi>d</mi></mstyle></math> +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s817" xml:space="preserve"> +Differentia +<lb/>[<emph style="it">tr: +Difference +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s818" xml:space="preserve"> +Operationes sunt <lb/> +in dorso D.1. +<lb/>[<emph style="it">tr: +The working is on the back of D.1. +</emph>]<lb/> +[<emph style="it">Note: +The back of sheet D.1 is Add MS 6782, f. 316v. + </emph>]<lb/> +</s> +</p> +<pb file="add_6782_f318v" o="318v" n="637"/> +<pb file="add_6782_f319" o="319" n="638"/> +<div xml:id="echoid-div213" type="page_commentary" level="2" n="213"> +<p> +<s xml:id="echoid-s819" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s819" xml:space="preserve"> +An examination of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>6</mn><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>2</mn><mi>a</mi><mo>=</mo><mn>8</mn></mstyle></math>, which has roots 2, 2, 2. +This is one of several equations with multiple roots treated by Viète in +<emph style="it">De potestatum numerosa resolutione</emph>. +Harriot solves it in full on Add MS 6783, f. 187, and refers to it again in Add MS 6783, f. 188. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head114" xml:space="preserve"> +G) +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s821" xml:space="preserve"> +Triens coeff: long: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> +<lb/>[<emph style="it">tr: +A third of the longitudinal coefficient +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s822" xml:space="preserve"> +Triplum quadrat: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>b</mi><mi>b</mi><mo>=</mo><mn>3</mn><mi>b</mi><mi>b</mi></mstyle></math> coeff. planis. +<lb/>[<emph style="it">tr: +Three times the square is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>b</mi><mi>b</mi><mn>4</mn><mo>,</mo><mi>t</mi><mi>h</mi><mi>e</mi><mi>p</mi><mi>l</mi><mi>a</mi><mi>n</mi><mi>e</mi><mi>c</mi><mi>o</mi><mi>e</mi><mi>f</mi><mi>f</mi><mi>i</mi><mi>c</mi><mi>i</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo>.</mo></mstyle></math></emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s823" xml:space="preserve"> +Duplus cubus e triente. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>b</mi><mi>b</mi><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Twice the cube of the third +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s824" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> <lb/> +in coeff: planib. +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> times the plane coefficient +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s825" xml:space="preserve"> +Tria latera igitur +<lb/>[<emph style="it">tr: +Therefore the three sides are +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f319v" o="319v" n="639"/> +<pb file="add_6782_f320" o="320" n="640"/> +<div xml:id="echoid-div214" type="page_commentary" level="2" n="214"> +<p> +<s xml:id="echoid-s826" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s826" xml:space="preserve"> +This set of pages, lettered <emph style="it">aa</emph> to <emph style="it">au</emph> is connected to +Harriot's treatise 'De generatione aequationum canonicarum' in Add MS 6783, f. 183 to f. 163. <lb/> +On this first page, Harriot works out the multiplications +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>c</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>c</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>c</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>a</mi><mi>a</mi><mo>-</mo><mi>b</mi><mi>b</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>a</mi><mi>a</mi><mo>-</mo><mi>b</mi><mi>b</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>a</mi><mi>a</mi><mo>+</mo><mi>b</mi><mi>b</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>. +In each case he writes down the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> that reduce the resulting product to zero. +For the second multiplication, for instance, he shows that the product becomes zero when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math> +but not when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>d</mi></mstyle></math>. <lb/> +For similar content see Add MS 6782, f. 182 (d.2), f. 181 (d.3), f. 180 (d.4), and f. 178 (d.6). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head115" xml:space="preserve"> +aa) +</head> +<pb file="add_6782_f320v" o="320v" n="641"/> +<pb file="add_6782_f321" o="321" n="642"/> +<div xml:id="echoid-div215" type="page_commentary" level="2" n="215"> +<p> +<s xml:id="echoid-s828" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s828" xml:space="preserve"> +This page contains multiplications similar to those on the previous page (Add MS 6782, f. 320) +but now with the terms written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>-</mo><mi>a</mi><mo maxsize="1">)</mo></mstyle></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo maxsize="1">)</mo></mstyle></math>, and so on. +Both versions are treated in Add MS 6782, f. 182 (d.2). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head116" xml:space="preserve"> +ab) +</head> +<pb file="add_6782_f321v" o="321v" n="643"/> +<pb file="add_6782_f322" o="322" n="644"/> +<head xml:id="echoid-head117" xml:space="preserve"> +ac) +</head> +<div xml:id="echoid-div216" type="page_commentary" level="2" n="216"> +<p> +<s xml:id="echoid-s830" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s830" xml:space="preserve"> +A treatment of the equation arising from the multiplication <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>c</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>, with a numerical example. +For a more detailed treatment of the same equation see Add MS 6782, f. 181 (d.3). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s832" xml:space="preserve"> +Fundamentum +<lb/>[<emph style="it">tr: +Foundation +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f322v" o="322v" n="645"/> +<pb file="add_6782_f323" o="323" n="646"/> +<div xml:id="echoid-div217" type="page_commentary" level="2" n="217"> +<p> +<s xml:id="echoid-s833" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s833" xml:space="preserve"> +A continuation from Add MS 6782, f. 322 of work on the equation arising from the multiplication +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>c</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>. <lb/> +Harriot states without proof the special form the equation will take when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>=</mo><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>, +when the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> vanishes. For a full derivation see Add MS 6782, f. 181 (d.3). +Harriot calls this form of the cubic equation an 'elliptic' or 'Bombellian' equation. +The special case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mi>c</mi></mstyle></math> he calls 'parabolic'. +For Harriot's definitions of the hyperbolic, elliptic, and parabolic forms +of a cubic equation without a square term, see Add MS 6783, f. 106 (e.8). <lb/> +On this page Harriot also gives the form the equation will take when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>d</mi><mo>+</mo><mi>c</mi><mi>d</mi></mstyle></math>, +when the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> vanishes. For a full derivation see Add MS 6782, f. 181 (d.3). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head118" xml:space="preserve"> +ad) +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s835" xml:space="preserve"> +In charta ac) +<lb/>[<emph style="it">tr: +In sheet <emph style="it">ac</emph> +</emph>]<lb/> +[<emph style="it">Note: +Sheet ac is Add MS 6782, f. 322. + </emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s836" xml:space="preserve"> +Eliptica. <lb/> +seu Bombellica si <lb/> +convertitur. +<lb/>[<emph style="it">tr: +Elliptic, or the Bombellian kind if the signs are changed. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s837" xml:space="preserve"> +Vide D.) +<lb/>[<emph style="it">tr: +See D.) +</emph>]<lb/> +[<emph style="it">Note: +Sheet D. is Add MS 6783, f. 272. + </emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s838" xml:space="preserve"> +æquatio parabolica. +<lb/>[<emph style="it">tr: +parabolic equation +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s839" xml:space="preserve"> +parabolica +<lb/>[<emph style="it">tr: +parabolic +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s840" xml:space="preserve"> +solummodo +<lb/>[<emph style="it">tr: +only +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f323v" o="323v" n="647"/> +<pb file="add_6782_f324" o="324" n="648"/> +<div xml:id="echoid-div218" type="page_commentary" level="2" n="218"> +<p> +<s xml:id="echoid-s841" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s841" xml:space="preserve"> +On this page Harriot works with the multiplication from Add MS 6782, f. 322 and f. 323, namely, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>c</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>, but now with the signs changed so that it becomes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>+</mo><mi>c</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>a</mi><mo>-</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>. +As in Add MS 6782, f. 323, he gives the special form of the equation that arises when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>=</mo><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>, +when the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> vanishes. For a full derivation see Add MS 6782, f. 180 (d.4). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head119" xml:space="preserve" xml:lang="lat"> +ae) Conversiones +<lb/>[<emph style="it">tr: +Changes of sign +</emph>]<lb/> +</head> +<pb file="add_6782_f324v" o="324v" n="649"/> +<pb file="add_6782_f325" o="325" n="650"/> +<pb file="add_6782_f325v" o="325v" n="651"/> +<head xml:id="echoid-head120" xml:space="preserve"> +f.8 +</head> +<pb file="add_6782_f326" o="326" n="652"/> +<pb file="add_6782_f326v" o="326v" n="653"/> +<head xml:id="echoid-head121" xml:space="preserve"> +f.8 +</head> +<pb file="add_6782_f327" o="327" n="654"/> +<pb file="add_6782_f327v" o="327v" n="655"/> +<head xml:id="echoid-head122" xml:space="preserve"> +f.8 +</head> +<pb file="add_6782_f328" o="328" n="656"/> +<pb file="add_6782_f328v" o="328v" n="657"/> +<pb file="add_6782_f329" o="329" n="658"/> +<pb file="add_6782_f329v" o="329v" n="659"/> +<p xml:lang="lat"> +<s xml:id="echoid-s843" xml:space="preserve"> +Archimedes de quadrat: parabola <lb/> +prop: 23. pa: 21. +<lb/>[<emph style="it">tr: +Archimedes, De quadratura parabola, Proposition 23, page 21. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s844" xml:space="preserve"> +decrescentes +<lb/>[<emph style="it">tr: +decreasing +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s845" xml:space="preserve"> +crescentes +<lb/>[<emph style="it">tr: +increasing +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f330" o="330" n="660"/> +<div xml:id="echoid-div219" type="page_commentary" level="2" n="219"> +<p> +<s xml:id="echoid-s846" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s846" xml:space="preserve"> +This page is a continuation of Add MS 6783, f. 44v. <lb/> +Note 3 gives the triangular numbers in general algebraic notation: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></mfrac></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo maxsize="1">)</mo></mrow><mrow><mn>1</mn><mo>×</mo><mn>2</mn></mrow></mfrac></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo maxsize="1">)</mo></mrow><mrow><mn>1</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>3</mn></mrow></mfrac></mstyle></math>, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>3</mn><mo maxsize="1">)</mo></mrow><mrow><mn>1</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>4</mn></mrow></mfrac></mstyle></math>. <lb/> +On the right these formula are given labels that in modern subscript notation would be +<emph style="it">p</emph><emph style="sub">1</emph>, <emph style="it">p</emph><emph style="sub">2</emph>, +<emph style="it">p</emph><emph style="sub">3</emph>, and so on. <lb/> +In the fourth notation, in the lower half of the page, +<emph style="it">p</emph> has been replaced by <emph style="it">v</emph>, +and the terms have been multiplied out to give a one-line expression (or in Harriot's terms, an equation) +instead of a fraction. <lb/> +See also page 1 of the 'Magisteria' (Add MS 6782, f. 108). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s848" xml:space="preserve"> +3. Generalis notatio <lb/> +triangularium <lb/> +in notis generalibus. +<lb/>[<emph style="it">tr: +3. General notation for triangular numbers in general symbols. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s849" xml:space="preserve"> +Melius ad continuam <lb/> +additionem triangularium. +<lb/>[<emph style="it">tr: +Better for continual addition of triangular numbers. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s850" xml:space="preserve"> +4. Quarta notatio <lb/> +per æquationes. +<lb/>[<emph style="it">tr: +4. Fourth notation, by means of equations. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f330v" o="330v" n="661"/> +<div xml:id="echoid-div220" type="page_commentary" level="2" n="220"> +<p> +<s xml:id="echoid-s851" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s851" xml:space="preserve"> +This page contains two magic squares. <lb/> +It also shows the digits 0 to 9 written in their Arabic form and in characters composed only of straight lines +(see also Add MS 6782, f. 30v). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f331" o="331" n="662"/> +<div xml:id="echoid-div221" type="page_commentary" level="2" n="221"> +<p> +<s xml:id="echoid-s853" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s853" xml:space="preserve"> +This appears to be the 'other paper' referred to on Add MS 6782, f. 38, +since the table at the top of this page is the same as the one that appears there. <lb/> +On the right, the first few entries from the third and fourth columns are written in factorial form, +showing why the ratios of the entries in the first two rows are 1 : 6 and 2 : 5. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head123" xml:space="preserve"> +Of combinations. +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s855" xml:space="preserve"> +Questi minuti numerator <lb/> +multiplicatur per 2. <lb/> +& denominator per 5. +<lb/>[<emph style="it">tr: +The numerator of these fractions is multiplied by 2 etc., the denominator by 5. +</emph>]<lb/> +</s> +<s xml:id="echoid-s856" xml:space="preserve"> +Ergo tertius et quartus <lb/> +habent ratione ut 2 ad 5. +<lb/>[<emph style="it">tr: +Therefore the third and the fourth have a ratio of 2 to 5. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s857" xml:space="preserve"> +unde ratio in omnibus. +<lb/>[<emph style="it">tr: +whence the ratio in all of them. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f331v" o="331v" n="663"/> +<pb file="add_6782_f332" o="332" n="664"/> +<div xml:id="echoid-div222" type="page_commentary" level="2" n="222"> +<p> +<s xml:id="echoid-s858" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s858" xml:space="preserve"> +A table of factorials from 1! = 1, to 25! = 15,511,210,043,330,985,984,000,000. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head124" xml:space="preserve"> +For Transpositions. +</head> +<pb file="add_6782_f332v" o="332v" n="665"/> +<pb file="add_6782_f333" o="333" n="666"/> +<pb file="add_6782_f333v" o="333v" n="667"/> +<pb file="add_6782_f334" o="334" n="668"/> +<pb file="add_6782_f334v" o="334v" n="669"/> +<pb file="add_6782_f335" o="335" n="670"/> +<div xml:id="echoid-div223" type="page_commentary" level="2" n="223"> +<p> +<s xml:id="echoid-s860" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s860" xml:space="preserve"> +The same information as in Add MS 6782, f. 336, now presented slightly differently. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head125" xml:lang="lat"> +Progressiones crescentes; quarum principia sunt quivis numeri. +<lb/>[<emph style="it">tr: +Increasing progressions; of which the first terms are any numbers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s862" xml:space="preserve"> +Melior forma <lb/> +sive optime. +<lb/>[<emph style="it">tr: +A better form, perhaps the best. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f335v" o="335v" n="671"/> +<pb file="add_6782_f336" o="336" n="672"/> +<div xml:id="echoid-div224" type="page_commentary" level="2" n="224"> +<p> +<s xml:id="echoid-s863" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s863" xml:space="preserve"> +General formulae for the entries in a table generated from a constant difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, +where every column is increasing (as signified by the symbols Δ above each column). <lb/> +The first entry in column 1 is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. <lb/> +The first entry in column 2 is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>2</mn></msup></mrow></mstyle></math> (where 2 is a superscript, not a power). <lb/> +The first entry in column 3 is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>3</mn></msup></mrow></mstyle></math> (where 3 is a superscript, not a power). <lb/> +And so on. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head126" xml:space="preserve" xml:lang="lat"> +progressiones crescentes, quarum principia <lb/> +sunt quibus numeri. +<lb/>[<emph style="it">tr: +increasing progressions, of which the first terms are any numbers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s865" xml:space="preserve"> +Vide meliorem <lb/> +formam in alia <lb/> +Charta. +<lb/>[<emph style="it">tr: +See a better form in the other sheet. +</emph>]<lb/> +<sc> +The other sheet mentioned here is Add MS 6782, f. 355. +</sc> +</s> +</p> +<pb file="add_6782_f336v" o="336v" n="673"/> +<pb file="add_6782_f337" o="337" n="674"/> +<div xml:id="echoid-div225" type="page_commentary" level="2" n="225"> +<p> +<s xml:id="echoid-s866" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s866" xml:space="preserve"> +These are the symbols Harriot devised in 1585 for writing down the native Indian language of Algonquin. +In the right-hand column there are 12 vowels followed by 24 consonants. +In the left-hand column are words representing each sound. +The sounds and then the words are transcribed here from Harriot's notes on another copy of this page, +now held at Westminster School, London, and reproduced in Stedall 2007. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p> +<s xml:id="echoid-s868" xml:space="preserve"> +[sounds] <lb/> +as (a) in all, tall, fall, call <lb/> +as (o) in ore, for, core <lb/> +as (a) in arrow, man, pan <lb/> +as (u) in us, upon, but, cut <lb/> +as (a) in ape, ale, any , are <lb/> +as (e) in erbe, end, the <lb/> +as (i) in ise, ire, pipe <lb/> +as (e) in he, shee, or (ee) in thee, eele. <lb/> +[penultimate pair of vowels] +(in barbarouse wordes only and not to be expressed viva voce.) <lb/> +as (o) in so, no, otes. <lb/> +as (o) in do, to, shoe <lb/> +as (y) in yea, yes, day <lb/> +as (w) in way, was, now, sow <lb/> +as (r) in roote <lb/> +as (l) in lake <lb/> +as (z) in zone, zachary <lb/> +as the French (i) in je, jeter, or as (g) in hodge, iudge <lb/> +as (s) in sault, samon <lb/> +as (sh) in she, shoe <lb/> +as (m) in man <lb/> +as (n) in not <lb/> +as (ng) in king, fling, thing (or (n) in knave) <lb/> +as (v) in vine, geve <lb/> +as (th) in the, thine, there <lb/> +as (gh) in some barabrouse wordes <lb/> +as (f) in fling, fear, of <lb/> +as (th) in thing, thorne <lb/> +as (ch) in some barbarouse wordes or as the Greeke <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>χ</mi></mstyle></math> <lb/> +as (h) in hat, he, oh <lb/> +as (b) in borde, but <lb/> +as (d) in do, grudge, good <lb/> +as (g) in good, god, geve, hog <lb/> +as (p) in pan <lb/> +as (t) in tooth, to, ten, hat <lb/> +as (c) in corne or as (k) in keepe +</s> +</p> +<p> +<s xml:id="echoid-s869" xml:space="preserve"> +[words] <lb/> +armes. ore. <lb/> +arow. urchin. <lb/> +aye. err. <lb/> +ice. eele. <lb/> +(in barbarous words only and not to be expressed viva voce.) <lb/> +oates. oon. <lb/> +ye. ne. <lb/> +root. lake. <lb/> +zone. je. <lb/> +sault. shoo. <lb/> +men. nete. gna. <lb/> +vine. thing. ghi. <lb/> +flinge. thorne. chi. <lb/> +bore. drudge. gold. <lb/> +pan. toothe corne. +</s> +</p> +<pb file="add_6782_f337v" o="337v" n="675"/> +<pb file="add_6782_f338" o="338" n="676"/> +<pb file="add_6782_f338v" o="338v" n="677"/> +<pb file="add_6782_f339" o="339" n="678"/> +<pb file="add_6782_f339v" o="339v" n="679"/> +<pb file="add_6782_f340" o="340" n="680"/> +<pb file="add_6782_f340v" o="340v" n="681"/> +<pb file="add_6782_f341" o="341" n="682"/> +<pb file="add_6782_f341v" o="341v" n="683"/> +<pb file="add_6782_f342" o="342" n="684"/> +<pb file="add_6782_f342v" o="342v" n="685"/> +<pb file="add_6782_f343" o="343" n="686"/> +<pb file="add_6782_f343v" o="343v" n="687"/> +<pb file="add_6782_f344" o="344" n="688"/> +<pb file="add_6782_f344v" o="344v" n="689"/> +<pb file="add_6782_f345" o="345" n="690"/> +<pb file="add_6782_f345v" o="345v" n="691"/> +<pb file="add_6782_f346" o="346" n="692"/> +<pb file="add_6782_f346v" o="346v" n="693"/> +<pb file="add_6782_f347" o="347" n="694"/> +<div xml:id="echoid-div226" type="page_commentary" level="2" n="226"> +<p> +<s xml:id="echoid-s870" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s870" xml:space="preserve"> +The table from page 12 of the 'Magisteria' (Add MS 6782, f. 119). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f347v" o="347v" n="695"/> +<pb file="add_6782_f348" o="348" n="696"/> +<div xml:id="echoid-div227" type="page_commentary" level="2" n="227"> +<p> +<s xml:id="echoid-s872" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s872" xml:space="preserve"> +See page 5 of the 'Magisteria' (Add MS 6782, f. 112), which contains the same numerical tables. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f348v" o="348v" n="697"/> +<div xml:id="echoid-div228" type="page_commentary" level="2" n="228"> +<p> +<s xml:id="echoid-s874" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s874" xml:space="preserve"> +See page 7 of the 'Magisteria' (Add MS 6782, f. 114), which contains the first two numerical tables. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f349" o="349" n="698"/> +<div xml:id="echoid-div229" type="page_commentary" level="2" n="229"> +<p> +<s xml:id="echoid-s876" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s876" xml:space="preserve"> +Formulae for entries in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> columns of a difference table; +see page 16 of the 'Magisteria' (Add MS 6782, f. 123). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f349v" o="349v" n="699"/> +<pb file="add_6782_f350" o="350" n="700"/> +<pb file="add_6782_f350v" o="350v" n="701"/> +<pb file="add_6782_f351" o="351" n="702"/> +<pb file="add_6782_f351v" o="351v" n="703"/> +<pb file="add_6782_f352" o="352" n="704"/> +<pb file="add_6782_f352v" o="352v" n="705"/> +<pb file="add_6782_f353" o="353" n="706"/> +<pb file="add_6782_f353v" o="353v" n="707"/> +<pb file="add_6782_f354" o="354" n="708"/> +<div xml:id="echoid-div230" type="page_commentary" level="2" n="230"> +<p> +<s xml:id="echoid-s878" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s878" xml:space="preserve"> +Squares of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo maxsize="1">)</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>+</mo><mi>f</mi><mo maxsize="1">)</mo></mstyle></math>. +The page also shows the calculation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>b</mi><mo>-</mo><mi>c</mi><mo maxsize="1">)</mo></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head127" xml:space="preserve" xml:lang="lat"> +Quadrata e polynomia radice +<lb/>[<emph style="it">tr: +Squares from polynomial roots. +</emph>]<lb/> +</head> +<pb file="add_6782_f354v" o="354v" n="709"/> +<pb file="add_6782_f355" o="355" n="710"/> +<div xml:id="echoid-div231" type="page_commentary" level="2" n="231"> +<p> +<s xml:id="echoid-s880" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s880" xml:space="preserve"> +Squares of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>–</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>–</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>–</mo><mi>c</mi><mo>–</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>, +and of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>–</mo><mi>f</mi><mo maxsize="1">)</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>–</mo><mi>d</mi><mo>–</mo><mi>f</mi><mo maxsize="1">)</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>–</mo><mi>c</mi><mo>–</mo><mi>d</mi><mo>–</mo><mi>f</mi><mo maxsize="1">)</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>–</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>–</mo><mi>f</mi><mo maxsize="1">)</mo></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head128" xml:space="preserve" xml:lang="lat"> +Quadrata e polynomia radice +<lb/>[<emph style="it">tr: +Squares from polynomial roots. +</emph>]<lb/> +</head> +<pb file="add_6782_f355v" o="355v" n="711"/> +<pb file="add_6782_f356" o="356" n="712"/> +<div xml:id="echoid-div232" type="page_commentary" level="2" n="232"> +<p> +<s xml:id="echoid-s882" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s882" xml:space="preserve"> +The square and cube of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f356v" o="356v" n="713"/> +<pb file="add_6782_f357" o="357" n="714"/> +<div xml:id="echoid-div233" type="page_commentary" level="2" n="233"> +<p> +<s xml:id="echoid-s884" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s884" xml:space="preserve"> +Powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo maxsize="1">)</mo></mstyle></math> up to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mrow><msup><mo maxsize="1">)</mo><mn>6</mn></msup></mrow></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head129" xml:space="preserve" xml:lang="lat"> +potentia e binomia radice +<lb/>[<emph style="it">tr: +powers from binomial roots +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s886" xml:space="preserve"> +solidum +<lb/>[<emph style="it">tr: +solid +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f357v" o="357v" n="715"/> +<pb file="add_6782_f358" o="358" n="716"/> +<div xml:id="echoid-div234" type="page_commentary" level="2" n="234"> +<p> +<s xml:id="echoid-s887" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s887" xml:space="preserve"> +Cubes of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>+</mo><mi>f</mi><mo maxsize="1">)</mo></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>+</mo><mi>f</mi><mo>+</mo><mi>g</mi><mo maxsize="1">)</mo></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<pb file="add_6782_f358v" o="358v" n="717"/> +<pb file="add_6782_f359" o="359" n="718"/> +<pb file="add_6782_f359v" o="359v" n="719"/> +<pb file="add_6782_f360" o="360" n="720"/> +<pb file="add_6782_f360v" o="360v" n="721"/> +<pb file="add_6782_f361" o="361" n="722"/> +<pb file="add_6782_f361v" o="361v" n="723"/> +<pb file="add_6782_f362" o="362" n="724"/> +<div xml:id="echoid-div235" type="page_commentary" level="2" n="235"> +<p> +<s xml:id="echoid-s889" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s889" xml:space="preserve"> +This page refers to Aristotle's <emph style="it">Physics</emph>, Books V and VI, +defines what it means for things to be together or apart, in contact, or continuous. +The definitions may be paraphrased as follows. +</s> +<lb/> +<quote> +Things are said to be together if they are in one place, apart if they are in different places. <lb/> +Things are said to be in contact if their extremities are together. <lb/> +Things are said to be continuous if the touching limits of each become one and the same. +</quote> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head130" xml:space="preserve" xml:lang="lat"> +De infinitis. De continuo. +<lb/>[<emph style="it">tr: +On infinity. On the continuum. +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s891" xml:space="preserve"> +Aristotle in the beginning of his 6th booke of his physicks, & in the <lb/> +26th treatise of the 5th booke, defineth those thinges to be <foreign xml:lang="lat">continua <lb/> +quorum extrema sunt unum.</foreign> And in the 22nd treatise of the said 5th booke <lb/> +that: <foreign xml:lang="lat">tangentia sunt, quorum extrema sunt simul</foreign>. +<foreign xml:lang="lat">Simul qua in <lb/> +uno loco sunt primo</foreign>. <foreign xml:lang="lat">Separatim qui sunt in altero.</foreign> +</s> +</p> +<p> +<s xml:id="echoid-s892" xml:space="preserve"> +Now for the <emph style="st">understanding</emph> +<emph style="super">better explication</emph> of the +<emph style="super">meaning of the</emph> definitions as also of their truth. Let us <lb/> +understand first two <emph style="super">materiall</emph> +cubes A & B to be separate, that is, to be in diverse <lb/> +planes, extremes & all. +</s> +</p> +<pb file="add_6782_f362v" o="362v" n="725"/> +<pb file="add_6782_f363" o="363" n="726"/> +<head xml:id="echoid-head131" xml:space="preserve" xml:lang="lat"> +De Infinitis progressionibus +<lb/>[<emph style="it">tr: +On infinite progressions +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s893" xml:space="preserve"> +In progressions that be infinite be they increasing or decreasing. <lb/> +</s> +<s xml:id="echoid-s894" xml:space="preserve"> +There are these passes. +</s> +<s xml:id="echoid-s895" xml:space="preserve"> +First to a quantity that haveth no +<emph style="st">proportion</emph> rate <lb/> +to the first quantity given, or rather because betwixt positive quantityes <lb/> +there is a positive rate, I may call that rate infinite either in great-<lb/> +ness or litle<emph style="super">nes</emph>s according to the +<emph style="st">proportion</emph> <emph style="super">progression</emph>, +in respect of the first quantity <lb/> +given. +</s> +<s xml:id="echoid-s896" xml:space="preserve"> +Yet in respecte of the progression following it is divisible or mul-<lb/> +tiplicable till the progression being infinite hath for his second passe <lb/> +also a quantity <emph style="super">of an[???]</emph> infinite rate. +</s> +<s xml:id="echoid-s897" xml:space="preserve"> +Which is not only infinite in respecte of <lb/> +the first quantity of the last progression; but infinitely infinite in respect <lb/> +of <emph style="st">of</emph> the first in the first progresse. +</s> +<s xml:id="echoid-s898" xml:space="preserve"> +And also the summe of the second pro-<lb/> +gression is infinite <emph style="st">infi</emph> +in respect of the first summe of the first pro-<lb/> +gression, or the first quantity of all. +</s> +</p> +<p> +<s xml:id="echoid-s899" xml:space="preserve"> +And so a third, fourth & infinite other progressions and passes; of which <lb/> +any quantity or the summe of all infinitely all, is of an infinite <lb/> +quantity in greatness of litleness in respect, of the summe or <lb/> +first quantity of the first progression. +</s> +<lb/> +<s xml:id="echoid-s900" xml:space="preserve"> +And yet <emph style="st">at</emph> <emph style="super">for a</emph> +last in decreasing progressions we must needes under-<lb/> +stand a quantity absolutely indivisible; but multiplicable infinitely <lb/> +infinite <emph style="st">to make the [¿]prime[?] +from where the rest are issued</emph> till a quantity <lb/> +absolutely immultiplicable be produced which I may call universally infinite. +</s> +<lb/> +<s xml:id="echoid-s901" xml:space="preserve"> +And in increasing progressions we must needes understand that <lb/> +<emph style="st">at</emph> <emph style="super">for a</emph> +last there must be a quantity immultiplicable absolute, but <lb/> +divisible infinitely infinite till that quantity be issued that is <lb/> +absolutely indivisble. +</s> +</p> +<p> +<s xml:id="echoid-s902" xml:space="preserve"> +That such <emph style="st">a</emph> quantity which I call universally infinite: hath not only <lb/> +act rationall, by supposition, or by consequence from +<emph style="super">mere</emph> supposition: but <lb/> +also act reall, or existence: in an instant, having +<emph style="super">[???] perfect</emph> actuall being, <lb/> +or in time, passed by motion <emph style="st">fini</emph> +both finite & infinite: with many reall <lb/> +consequences or properties consequent; & accidents adioyning: <lb/> +shalbe declared in the papers following. +</s> +</p> +<pb file="add_6782_f363v" o="363v" n="727"/> +<pb file="add_6782_f364" o="364" n="728"/> +<head xml:id="echoid-head132" xml:space="preserve" xml:lang="lat"> +De Infinitis. +<lb/>[<emph style="it">tr: +On infinity +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s903" xml:space="preserve"> +Seing that any finite line will <lb/> +subtend an angle at summe distance; <lb/> +as let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> subtend the <emph style="st">the</emph> angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi><mi>c</mi></mstyle></math>. +</s> +<lb/> +<s xml:id="echoid-s904" xml:space="preserve"> +Then a line double to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, which let be <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math>, will subtend the same angle at a <lb/> +double distance, so that <emph style="it"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math></emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi></mstyle></math> will be <lb/> +aequall to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. +</s> +</p> +<p> +<s xml:id="echoid-s905" xml:space="preserve"> +In those subtensions I understand that the poynt <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be <emph style="super">in a</emph> +perpendicular <emph style="super">line</emph> to the <lb/> +middle of the subtendent lines. +</s> +<s xml:id="echoid-s906" xml:space="preserve"> +as also in all the others which follow. +</s> +</p> +<p> +<s xml:id="echoid-s907" xml:space="preserve"> +Now I suppose <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> to be removed to a further distance from the poynt <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/> +</s> +<s xml:id="echoid-s908" xml:space="preserve"> +Then the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi><mi>c</mi></mstyle></math> subtended must be lesse than before. +</s> +<s xml:id="echoid-s909" xml:space="preserve"> +And <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math>. <lb/> +shall <emph style="st">[???]</emph> subtend the same angle at a double distance as before. +</s> +</p> +<p> +<s xml:id="echoid-s910" xml:space="preserve"> +And this is true <emph style="st">generally</emph> continually that the further +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> is removed <lb/> +the lesse angle it subtendeth & <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math> always must subtend the same <lb/> +angle at a double distance. +</s> +</p> +<p> +<s xml:id="echoid-s911" xml:space="preserve"> +Then I suppose <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> to be removed to an infinite distance; at which <lb/> +distance the supposition altereth not the quantity of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. but the +<emph style="st">quantity </emph>consequence <lb/> +is of the angle. +</s> +<s xml:id="echoid-s912" xml:space="preserve"> +Which wilbe, that the angle <emph style="st">wh</emph> +then subtended <emph style="it">[???]</emph> to be <lb/> +of an infinite quantity in litleness in respecte of the former angles. +</s> +<s xml:id="echoid-s913" xml:space="preserve"> +Yet it <lb/> +cannot be sayd to be no angle negatively because it is positive. & it <lb/> +must also follow that the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math> must subtend the same positive angle <lb/> +at a double distance. +</s> +<s xml:id="echoid-s914" xml:space="preserve"> +Which is Double to the former infinite distance. +</s> +</p> +<p> +<s xml:id="echoid-s915" xml:space="preserve"> +Also, let the distance of the subtendents be nearer <emph style="st">[???]</emph> +<emph style="super">to</emph> infinite, +<emph style="st">[???]</emph>it cannot be <lb/> +otherwise inferred but that the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math> & <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>g</mi></mstyle></math> +<emph style="st">being infinit</emph> though infinite, <lb/> +be <foreign xml:lang="lat ">ad diversas partes, & in diversis locis</foreign>, +because <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> & <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math> are betweene them, <lb/> +& have agreement or concurrence but only in the poynt <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, +<emph style="st">[???]</emph> <emph style="super">or</emph> in no distance <lb/> +out of the poynt <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. +</s> +</p> +<p> +<s xml:id="echoid-s916" xml:space="preserve"> +And yet the nearness of there congruence & +con<emph style="super">cu</emph>rrence in all other partes <lb/> +[???] at the utmost is such, that although they be remote; the angle <lb/> +is of no proportion explicable by nomber finite, but infinite +[¿]unknown[?], to any <lb/> +<emph style="st">angles</emph> other angle which we call finite. +</s> +<s xml:id="echoid-s917" xml:space="preserve"> +The like inexplicable proportion <lb/> +is of the <emph style="super">subtendent</emph> +lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math> & <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, to there infinite distance position from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. +</s> +<lb/> +<s xml:id="echoid-s918" xml:space="preserve"> +And yet the sayd lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math> & <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. as also that infinite litle or improportio-<lb/> +nable angle is divisible still <foreign xml:lang="lat">in infinitum</foreign>. & +still, although improportionable <lb/> +yet in an other respect, that is to say of his owne partes, is proportionable. +</s> +</p> +<pb file="add_6782_f364v" o="364v" n="729"/> +<pb file="add_6782_f365" o="365" n="730"/> +<head xml:id="echoid-head133" xml:space="preserve" xml:lang="lat"> +De Infinitis. +<lb/>[<emph style="it">tr: +On infinity +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s919" xml:space="preserve"> +That in a finite time an infinite space <lb/> +may be moved +</s> +</p> +<p> +<s xml:id="echoid-s920" xml:space="preserve"> +It is now convenient <lb/> +that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>f</mi></mstyle></math> be in this line. +</s> +</p> +<p> +<s xml:id="echoid-s921" xml:space="preserve"> +Suppose the <emph style="st">the</emph> line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>e</mi><mi>f</mi></mstyle></math> <foreign xml:lang="lat">et ultra</foreign> <lb/> +to be infinite, & the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> suppose <lb/> +to revolve & describe a circle <lb/> +in a finite time, fro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> towards <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> doth first respect <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <lb/> +after <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, & so forth successively no poynt <lb/> +in the infinite line is <emph style="st">not</emph> +unrespected by that time the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> cometh <lb/> +to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>g</mi></mstyle></math> where then the line is parallel & cutteth not the former <lb/> +line infinite. +</s> +<s xml:id="echoid-s922" xml:space="preserve"> +Now seing that a motion may be of any thing <lb/> +according <emph style="super">to</emph> the continuall succession of a poynt, +as well in respect <lb/> +of <emph style="st">[???]</emph> +<foreign xml:lang="lat">mobile ab motus</foreign>. +</s> +<s xml:id="echoid-s923" xml:space="preserve"> +Whatsoever may be or not be in <lb/> +respect of the moment, it maketh no matter: the purpose is <lb/> +manifest. +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s924" xml:space="preserve"> +Consequentia <lb/> +Accidentis quædam huius motus. +</s> +</p> +<p> +<s xml:id="echoid-s925" xml:space="preserve"> +The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> having moved till he comes to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math> that is <lb/> +parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>f</mi></mstyle></math>. & so that continuing his motion of revolution: +</s> +</p> +<p> +<s xml:id="echoid-s926" xml:space="preserve"> +The lines are parallel but in one instant. +</s> +<lb/> +<s xml:id="echoid-s927" xml:space="preserve"> +They never cut at an infinite distance but at that instant <lb/> +they are parallel. +</s> +<lb/> +<s xml:id="echoid-s928" xml:space="preserve"> +And if they cut then, they must cut +<foreign xml:lang="lat">ad utrasque partes</foreign> & then <lb/> +being right lines there must be no space betwixte them, but <lb/> +there distance by supposition is more than the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. +</s> +<s xml:id="echoid-s929" xml:space="preserve"> +Which <lb/> +implies contradiction. +</s> +</p> +<p> +<s xml:id="echoid-s930" xml:space="preserve"> +And yet there must be a cutting at an infinite distance or else all <lb/> +the poyntes of the infinite line could not have been respected. & +</s> +<s xml:id="echoid-s931" xml:space="preserve"> +if <lb/> +that be not some part of the infinite line, +that is some quantity <emph style="st">which</emph> <lb/> +<emph style="st">it is</emph> finite is only cut; +& that is at a finite distance; & then it maketh an <lb/> +angle <emph style="super">of quantity</emph> +at the greatest distance of such cutting: from that cutting the line <lb/> +by motion came to be parallel: That motion is made in an instant or <lb/> +in time. +</s> +<s xml:id="echoid-s932" xml:space="preserve"> +If in time, then in half the time the cutting must be further <lb/> +than the supposed furthest; +</s> +<s xml:id="echoid-s933" xml:space="preserve"> +If in an instant, our line wilbe in <emph style="st">two places</emph> <lb/> +two places in one <emph style="st">[???]</emph> instant; +<foreign xml:lang="lat">quæ implicant</foreign>. +</s> +</p> +<p> +<s xml:id="echoid-s934" xml:space="preserve"> +The lines therefore must cut at an infinite distance before they come to <lb/> +be parallel. +</s> +<s xml:id="echoid-s935" xml:space="preserve"> +And that must be in time before or in an instant before. <lb/> +</s> +<s xml:id="echoid-s936" xml:space="preserve"> +If in time, then in half the time they cut at greater distance than infinite or <lb/> +are parallel before they are parallel. +</s> +<s xml:id="echoid-s937" xml:space="preserve"> +Which both do imply contradiction. <lb/> +</s> +<s xml:id="echoid-s938" xml:space="preserve"> +If in an instant before; the two instants are one or different. +</s> +<s xml:id="echoid-s939" xml:space="preserve"> +If one, <foreign xml:lang="lat">implicat</foreign>. <lb/> +</s> +<s xml:id="echoid-s940" xml:space="preserve"> +If two there must be no other betwixt them. +</s> +<s xml:id="echoid-s941" xml:space="preserve"> +And then there [???] be a time greater <lb/> +than an instant & lesse than any time of quantity that is indivisible, that is <lb/> +agayne, indivisible into partes of quantity. & so also like of poyntes &c. +</s> +</p> +<pb file="add_6782_f365v" o="365v" n="731"/> +<pb file="add_6782_f366" o="366" n="732"/> +<head xml:id="echoid-head134" xml:space="preserve" xml:lang="lat"> +De Infinitis. +<lb/>[<emph style="it">tr: +On infinity +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s942" xml:space="preserve"> +<emph style="st">If the</emph> <emph style="super">The</emph> line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> by his revolution <lb/> +cometh at length to be parallel to <lb/> +the infinite line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math>. +</s> +<s xml:id="echoid-s943" xml:space="preserve"> +Which <lb/> +motion being from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> suppose <lb/> +to have been æqually. +</s> +<s xml:id="echoid-s944" xml:space="preserve"> +The <lb/> +degree of the motion let be <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>n</mi></mstyle></math>. the time <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>p</mi></mstyle></math>. +</s> +<s xml:id="echoid-s945" xml:space="preserve"> +The beginning of the time or first instant <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. <lb/> +</s> +<s xml:id="echoid-s946" xml:space="preserve"> +The last instant wherein the line is <lb/> +parallel, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. Now seing that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> must cut at <lb/> +an infinite distance & <emph style="super">that</emph> his last cutting must be <lb/> +before the instant <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. +</s> +<s xml:id="echoid-s947" xml:space="preserve"> +Which suppose <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math>. +</s> +<s xml:id="echoid-s948" xml:space="preserve"> +That <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math> as it is argued by the premises <lb/> +must differe from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> by an indivisible time, so that <emph style="st">it </emph> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math> must be the next instant <lb/> +to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. & no other between. +</s> +<s xml:id="echoid-s949" xml:space="preserve"> +In which instant <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> must not be parallel but <lb/> +make his last cutting at an infinite distance. +</s> +<s xml:id="echoid-s950" xml:space="preserve"> +And therefore it must have <lb/> +a certayne <foreign xml:lang="lat">situs</foreign> +at that instant out of the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> towards <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, which let be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math>, <lb/> +as it maketh his last section. +</s> +<s xml:id="echoid-s951" xml:space="preserve"> +In which situation the motion ordering it hath <lb/> +the sayd degree <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>n</mi></mstyle></math>, as in all other situations. +</s> +<s xml:id="echoid-s952" xml:space="preserve"> +From the which situation to the situation <lb/> +of being parallel it must be moved unto (as it is sayd) in the next instant. +</s> +</p> +<p> +<s xml:id="echoid-s953" xml:space="preserve"> +Now suppose (as it may be) that the motion from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> be in half the time <lb/> +of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>p</mi></mstyle></math>. +</s> +<s xml:id="echoid-s954" xml:space="preserve"> +Then doth it follow necessarily that the degree of motion or velo-<lb/> +city be double to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>n</mi></mstyle></math>. And therefore, what space or parte of a space, (be it <lb/> +finite or infinite, so it be positive,) it moved before according to <lb/> +the degree of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>n</mi></mstyle></math>. it moveth the same now, in half the time. <lb/> +</s> +<s xml:id="echoid-s955" xml:space="preserve"> +Therefore in this second motion when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> cometh to have his situation <lb/> +at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math> to make the sayd last section; seing that then it hath double <lb/> +degree of velocity; it must afterward be parallel in half an instant <lb/> +that is to say, <emph style="st">that</emph> +in half that time which was sayd to be indivisible. <lb/> +</s> +<s xml:id="echoid-s956" xml:space="preserve"> +Which doth imply contradiction. +</s> +</p> +<p> +<s xml:id="echoid-s957" xml:space="preserve"> +Agayne if it be sayd that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math> at that instant +<emph style="st">& in the position</emph> (when & <lb/> +where it maketh his last section with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> before it be parallel) +<emph style="st">then</emph> <lb/> +be <foreign xml:lang="lat">deinceps</foreign> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>. or that the poynts <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> & <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> be +<foreign xml:lang="lat">deinceps</foreign> at an infinite <lb/> +distance so that no point can be between. +</s> +<s xml:id="echoid-s958" xml:space="preserve"> +Yet from the poynt <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> may <lb/> +be interposed a line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>f</mi></mstyle></math>. and also from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. & by the doctrine of Elements <lb/> +the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>k</mi><mi>h</mi></mstyle></math>, or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>l</mi><mi>h</mi></mstyle></math> must be +<emph style="st">greater</emph> <emph style="super">lesser</emph> +than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>a</mi><mi>h</mi></mstyle></math>. & therefore lesse than that <lb/> +which was sayd to be least or indivisible. & therefore the lines <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>a</mi><mi>h</mi></mstyle></math>, or the <lb/> +poynts <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> & <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> be not <foreign xml:lang="lat">deinceps. quæ implicant</foreign>. +</s> +</p> +<pb file="add_6782_f366v" o="366v" n="733"/> +<pb file="add_6782_f367" o="367" n="734"/> +<head xml:id="echoid-head135" xml:space="preserve" xml:lang="lat"> +De Infinitis. Ratio Achilles +<lb/>[<emph style="it">tr: +On infinity. The ratio of Achilles +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s959" xml:space="preserve"> +There is a reason <emph style="super">of Zeno</emph> in Aristotle +(in the 6th booke of his phisickes. text. 78.) which <lb/> +for the sorce it seemeth to carry is called Achilles. +</s> +<s xml:id="echoid-s960" xml:space="preserve"> +And for that cause, no doubt, <lb/> +is <emph style="super">the</emph> name also Achilles used in the example to expresse the reason. +</s> +<s xml:id="echoid-s961" xml:space="preserve"> +The which <lb/> +because it is against Aristotles doctrine +& for that it compryseth matter <emph style="super">pregnant</emph> <lb/> +of greater consequence concerning the doctrine of infinites, it being there <lb/> +but briefly & obscurely set downe with an answere uncertayne: I thinke good <lb/> +to set <emph style="st">[???]</emph>downe more [???] & largely: +with Aristotles Answere as he hath <lb/> +it in the place allwayes, as also at full according to his owne doctrine in <lb/> +other places. +</s> +<s xml:id="echoid-s962" xml:space="preserve"> +To the end that comparing one with the other, the truth may appear, <lb/> +& perhaps [¿]seem[?] otherwise to be, +then yet hath been by the peripateticles either noted or <lb/> +observed. +</s> +</p> +<p> +<s xml:id="echoid-s963" xml:space="preserve"> +The proposition of Zeno is. +</s> +<s xml:id="echoid-s964" xml:space="preserve"> +The swift runner (runne he never so <lb/> +swiftly) shall never overtake the slow runner <emph style="super">mover</emph> +(runne <emph style="super">move</emph> he never <lb/> +so slowly. +</s> +<lb/> +<s xml:id="echoid-s965" xml:space="preserve"> +That there may be <emph style="super">no</emph> doubte of the meaning of the <lb/> +proposition we will declare what thinges are therein supposed. +</s> +<lb/> +<s xml:id="echoid-s966" xml:space="preserve"> +<emph style="st">The suppositions for the reason are adjoyned.</emph> +</s> +<lb/> +<s xml:id="echoid-s967" xml:space="preserve"> +ffirst, (as it ought to be, else the proposition were ridiculous) The motion <lb/> +of the runner & slow mover are understood to be both one way & in <lb/> +one right line. +</s> +<lb/> +<s xml:id="echoid-s968" xml:space="preserve"> +Secondly <emph style="st">the [???] of [???] must be of some [???]</emph> +The <lb/> +</s> +</p> +<pb file="add_6782_f367v" o="367v" n="735"/> +<pb file="add_6782_f368" o="368" n="736"/> +<head xml:id="echoid-head136" xml:space="preserve" xml:lang="lat"> +Ratio Achilles. +<lb/>[<emph style="it">tr: +The ratio of Achilles +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s969" xml:space="preserve"> +Let Achilles be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>. +</s> +<lb/> +<s xml:id="echoid-s970" xml:space="preserve"> +Testudo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +The tortoise <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>. +</emph>]<lb/> +</s> +<s xml:id="echoid-s971" xml:space="preserve"> +The Motion of Achilles from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> <lb/> +in the time <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math>. of Testudo from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math> <lb/> +in the <emph style="super">same</emph> time <emph style="st">fg</emph>. +</s> +<s xml:id="echoid-s972" xml:space="preserve"> +<emph style="st">Which let be the</emph> <lb/> +<emph style="st">half parte of the time ef</emph>. +</s> +<s xml:id="echoid-s973" xml:space="preserve"> +Which <lb/> +space of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math> let be the tenth parte <lb/> +of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>. +</s> +<s xml:id="echoid-s974" xml:space="preserve"> +Now the quaestion is, <lb/> +both these motions being continued in the same proportion as 10 to 1. <lb/> +where & when shall <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> overtake <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>. <emph style="st">Suppose at d</emph>. <lb/> +</s> +<s xml:id="echoid-s975" xml:space="preserve"> +At some point or other it must really be. +</s> +<s xml:id="echoid-s976" xml:space="preserve"> +Suppose that <emph style="super">X</emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. +</s> +<s xml:id="echoid-s977" xml:space="preserve"> +There must be <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> & <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>, <emph style="st">[???]</emph> +at the same instant of time. +</s> +<s xml:id="echoid-s978" xml:space="preserve"> +And therefore the time wherein <lb/> +<emph style="st">that</emph> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> hath moved to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> must be the same wherein <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> hath moved to <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. +</s> +<s xml:id="echoid-s979" xml:space="preserve"> +But the space <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>d</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>d</mi></mstyle></math> must be as 10 to 1. +</s> +<lb/> +<s xml:id="echoid-s980" xml:space="preserve"> +Now by the supposition it must follow +(because these motions be proportionall <emph style="super">(as 10 to 1)</emph>) <lb/> +* As <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math>. so: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>d</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>d</mi></mstyle></math>. which same termes <lb/> +proportionall call by these <emph style="st">same</emph> letters & in the same order. +</s> +</p> +<p> +<s xml:id="echoid-s981" xml:space="preserve"> +As <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math> is known to be 1. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>γ</mi></mstyle></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mstyle></math>. +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mo>+</mo><mi>α</mi></mstyle></math> is unknown. & so is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi></mstyle></math>. <lb/> +yet this is known that. <emph style="st">is æquall to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ɛ</mi></mstyle></math>.</emph> +</s> +</p> +<p> +<s xml:id="echoid-s982" xml:space="preserve"> +* Now what other proportion is this than if a man <lb/> +should say as <emph style="st">all</emph> the first to the second so <lb/> +all the antecedents to all the consequents which <lb/> +in this be infinite in nomber. +</s> +</p> +<p> +<s xml:id="echoid-s983" xml:space="preserve"> +X To find that poynt geometrically is set downe <lb/> +in my other papers <foreign xml:lang="lat">de infinitis</foreign>. +<lb/>[<emph style="it">tr: +on infinity +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f368v" o="368v" n="737"/> +<pb file="add_6782_f369" o="369" n="738"/> +<head xml:id="echoid-head137" xml:space="preserve" xml:lang="lat"> +De Infinitis. +<lb/>[<emph style="it">tr: +On infinity +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s984" xml:space="preserve"> +Now will I propound some dfficultyes to be <lb/> +considered of. +</s> +<s xml:id="echoid-s985" xml:space="preserve"> +Seing that every line is compounded <lb/> +of atomes, & therefore the periphery of a circle. <emph style="st">that <lb/> +is to say</emph> one +<foreign xml:lang="lat">atomus</foreign> is succeeding one an other <lb/> +infinitely in such manner as <emph style="it">that</emph> the perifery is at <lb/> +last compounded and made. +</s> +</p> +<p> +<s xml:id="echoid-s986" xml:space="preserve"> +Now also seing that the whole <foreign xml:lang="lat">periferies</foreign> +is compounded of <foreign xml:lang="lat">atomis undiquaque <lb/> +sitis</foreign> about the poynt <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. so many times infinitely, & to that number of them <lb/> +infinitely, till the circle supposed be accomplished. +</s> +</p> +<p> +<s xml:id="echoid-s987" xml:space="preserve"> +I demand <emph style="st">therefore</emph> <emph style="super">then</emph> +what wilbe the nomber of <foreign xml:lang="lat">atomi</foreign> +that are <foreign xml:lang="lat">deinceps</foreign> about the <lb/> +point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. +</s> +<s xml:id="echoid-s988" xml:space="preserve"> +Infinite they must needes be, or else infinite lines could not <lb/> +be <emph style="st">dra</emph> supposed actually from the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> to the perifery. +</s> +<s xml:id="echoid-s989" xml:space="preserve"> +And infinite also <lb/> +are <emph style="super">also</emph> in the perifery. +</s> +<s xml:id="echoid-s990" xml:space="preserve"> +But now I demande whether they are aequally infinite <lb/> +or not. +</s> +<s xml:id="echoid-s991" xml:space="preserve"> +If about the center are lesse infinite then there cannot from the <lb/> +center <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> to every poynt in the perifery be understood a right line but <lb/> +we must understand those <emph style="super"><foreign xml:lang="lat">atomi</foreign> about the center</emph> +that we supposed indivisible, divisble <emph style="super">which were absurd</emph>. +</s> +<s xml:id="echoid-s992" xml:space="preserve"> +and <lb/> +if they be æqually infinite: then <emph style="st">the same nomber of +<foreign xml:lang="lat">atomi</foreign></emph> in a great <lb/> +place, (where the nomber, although infinite, yet in them selves definite; because <lb/> +they being supposed to have <emph style="st">[???]</emph> +acte there is not one more nor lesse. +</s> +<s xml:id="echoid-s993" xml:space="preserve"> +Neither <lb/> +can there be more because <emph style="st">[???]</emph> +they being <foreign xml:lang="lat">deinceps</foreign> one more cannot <lb/> +be between there being no distance: & if there <emph style="st">be supposed</emph> +<emph style="super">might be</emph> one lesse; there <lb/> +lacketh of the supposed actaull, & definite & positive number although infinite. <lb/> +</s> +<s xml:id="echoid-s994" xml:space="preserve"> +Then I say in a greate place where there could be no more or lesse, <lb/> +in a lesse place there are an æquall nomber; which seemeth to imply. +</s> +</p> +<p> +<s xml:id="echoid-s995" xml:space="preserve"> +An other difficulty riseth from the square. +</s> +<s xml:id="echoid-s996" xml:space="preserve"> +If a line <lb/> +be compounded of <foreign xml:lang="lat">atomis</foreign>, the diametrall line wilbe <lb/> +found to be aæquall to the side. +</s> +<s xml:id="echoid-s997" xml:space="preserve"> +ffor suppose the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> <lb/> +to be drawne <emph style="st">from the poynt[???]</emph> +from the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> <emph style="super">of the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math>,</emph> to <lb/> +the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, of <emph style="st">th</emph> the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. Then from the next point <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, which is <foreign xml:lang="lat">deinceps</foreign> +to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> in the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math>, draw a line <lb/> +to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> the next point to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> in the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. +</s> +<s xml:id="echoid-s998" xml:space="preserve"> +So likewise from every <lb/> +next point in the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math>, to every next point in the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. <lb/> +</s> +<s xml:id="echoid-s999" xml:space="preserve"> +Now the lines so drawne must needs be the least & most that may be, <lb/> +because they are <foreign xml:lang="lat">deinceps</foreign> & all. +& they all cut the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> & of the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> <lb/> +there can be <emph style="st">between</emph> no point betwixt +<emph style="st">the</emph> two of the former lines +<emph style="super">because they are +<foreign xml:lang="lat">deinceps</foreign></emph>. +</s> +<s xml:id="echoid-s1000" xml:space="preserve"> +And therefore <lb/> +the nomber of the poynts of the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>, +<emph style="super">are</emph> aequally infinite to the poynts of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> & <lb/> +<foreign xml:lang="lat">per</foreign> consequence the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> & <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> aequall. +</s> +<s xml:id="echoid-s1001" xml:space="preserve"> +But this difficulty wilbe made more <lb/> +playne by the next following, which <emph style="st">[???]</emph> +wilbe found the meanes for the solution <lb/> +of all. +</s> +</p> +<p> +<s xml:id="echoid-s1002" xml:space="preserve"> +An other question is. where two +<foreign xml:lang="lat">atomi</foreign> are +<foreign xml:lang="lat">deinceps</foreign>. whether an other <lb/> +(the <emph style="st">other</emph> <emph style="super">first two</emph> +not disioyned) may either passe or have situation betwixt them. +</s> +</p> +<pb file="add_6782_f369v" o="369v" n="739"/> +<pb file="add_6782_f370" o="370" n="740"/> +<head xml:id="echoid-head138" xml:space="preserve" xml:lang="lat"> +De Infinitis. Notanda. +<lb/>[<emph style="it">tr: +On infinity. To be noted. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1003" xml:space="preserve"> +De tactu duorum corporum <lb/> +per superficies. an duæ <lb/> +superficies sint realiter distantes <lb/> +in corporum contactu. +<lb/>[<emph style="it">tr: +On the contact of two bodies at their surfaces, +but the two surfaces are in reality separate in the contact of the bodies. +</emph>]<lb/> +</s> +</p> +<p> +<s xml:id="echoid-s1004" xml:space="preserve"> +Because <foreign xml:lang="lat">discretum</foreign> +is negative to <foreign xml:lang="lat">continuum</foreign> <lb/> +in respect of that thing <emph style="st">that</emph> +<emph style="super">which</emph> may be sayd to be <lb/> +either. +</s> +<s xml:id="echoid-s1005" xml:space="preserve"> +If yet <emph style="super">that</emph> +which is <foreign xml:lang="lat">discretum</foreign> +is not <foreign xml:lang="lat">continuum</foreign> <lb/> +& that which is <foreign xml:lang="lat">continuum</foreign> +is not <foreign xml:lang="lat">discretum</foreign>. therefore <lb/> +the one being knowne the other cannot be <lb/> +unknowne what it is. +</s> +<s xml:id="echoid-s1006" xml:space="preserve"> +Now although there <lb/> +be great controversy of the essence & quality <lb/> +of <foreign xml:lang="lat">continuum</foreign>. +yet there is no such of <foreign xml:lang="lat">discretum</foreign>. <lb/> +we will therefore lay downe what is manifest <lb/> +of it, that the ratio & essence of <foreign xml:lang="lat">continuum</foreign> may appeare. +</s> +</p> +<pb file="add_6782_f370v" o="370v" n="741"/> +<p> +<s xml:id="echoid-s1007" xml:space="preserve"> +Willaim Sprat a wolle draper <lb/> +at the sign of the rope in Watlin <lb/> +street at Soper Lane corner. serveth <lb/> +for a [???] for his wifes brother <lb/> +for [???]. there are 4. +</s> +</p> +<pb file="add_6782_f371" o="371" n="742"/> +<head xml:id="echoid-head139" xml:space="preserve" xml:lang="lat"> +De Infinitis. +<lb/>[<emph style="it">tr: +On infinity +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s1008" xml:space="preserve"> +That there may be two magnitudes given, of which the one shalbe <lb/> +infinite in respect of the other, & yet in respect of two other <lb/> +magnitudes they shalbe finite. +</s> +</p> +<p> +<s xml:id="echoid-s1009" xml:space="preserve"> +That a line finite, cannot have his partes, of a finite magnitude; but <lb/> +they must be of a finite nomber. +</s> +</p> +<p> +<s xml:id="echoid-s1010" xml:space="preserve"> +That a finite line may have an infinite nomber of partes, +& if <emph style="super">all</emph> the <lb/> +partes be in continuall proportion: the nomber must be compounded <lb/> +of an infinite nomber of finite partes; & an infinite nomber of <lb/> +infinite partes. +</s> +</p> +<p> +<s xml:id="echoid-s1011" xml:space="preserve"> +If a line be understood to be compounded of +<emph style="st">infinite</emph> poyntes: <lb/> +the nomber of them is infinite of the first passe, second or <lb/> +any nomber of passes finite or infinite. +</s> +</p> +<pb file="add_6782_f371v" o="371v" n="743"/> +<pb file="add_6782_f372" o="372" n="744"/> +<head xml:id="echoid-head140" xml:space="preserve" xml:lang="lat"> +De Infinitis. Ratio Clava Herculis. +<lb/>[<emph style="it">tr: +On infinity. The ratio of the key of Hercules +</emph>]<lb/> +</head> +<pb file="add_6782_f372v" o="372v" n="745"/> +<pb file="add_6782_f373" o="373" n="746"/> +<head xml:id="echoid-head141" xml:space="preserve" xml:lang="lat"> +De Infinitis. +<lb/>[<emph style="it">tr: +On infinity +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s1012" xml:space="preserve"> +Suppose the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> doth touch the <lb/> +the circle in the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. & touching <lb/> +in that point <emph style="super">in</emph> it only & in no other <lb/> +point <emph style="st">it</emph> +<emph style="super">it</emph> toucheth, as Euclide suffi-<lb/> +ciently demonstrateth. Now I say <lb/> +there is (a point <foreign xml:lang="lat">deinceps</foreign>) a next <lb/> +poynt that doth not touch the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. +</s> +</p> +<pb file="add_6782_f373v" o="373v" n="747"/> +<pb file="add_6782_f374" o="374" n="748"/> +<head xml:id="echoid-head142" xml:space="preserve" xml:lang="lat"> +De Infinitis. +<lb/>[<emph style="it">tr: +On infinity +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s1013" xml:space="preserve"> +Minimum. That will kill men by <lb/> +piercing & running through. <lb/> +</s> +<s xml:id="echoid-s1014" xml:space="preserve"> +Maximum. That which will presse men <lb/> +to death. +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1015" xml:space="preserve"> +Unitas. Numeris unitatum. <lb/> +finitis <lb/> +infinitis +<lb/>[<emph style="it">tr: +Unity. The number of a unit. Finite. Infinite. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1016" xml:space="preserve"> +Finites finitorum. <lb/> +Infinites finitorum. <lb/> +finites Infinitorum. <lb/> +Infinites Infinitorum. <lb/> +Infiniti infinitorum infinitum. <lb/> +Infiniti infinitorum finitum. +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1017" xml:space="preserve"> +finitorum minimum. <lb/> +Infinitorum minimum. <lb/> +finitus minimorum. <lb/> +Infinites minimorum. <lb/> +finites finiti minimorum. <lb/> +Infinites finiti minimorum. <lb/> +Infinites finiti maximorum. <lb/> +Infinites infiniti maximorum. <lb/> +finiti. <lb/> +finitorum maximum .1. Infinitorum <lb/> +Infinitorum maximum. <lb/> +Infiniti. +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1018" xml:space="preserve"> +Ratio Achilles +<lb/>[<emph style="it">tr: +The ratio of Achilles +</emph>]<lb/> +</s> +</p> +<p> +<s xml:id="echoid-s1019" xml:space="preserve"> +All the mistery of infinites lieth <lb/> +in <foreign xml:lang="lat">formati ratione +<emph style="super">unius</emph> unitatis</foreign> <lb/> +which is only respective, & from <lb/> +where the knowledge & import <lb/> +of <foreign xml:lang="lat">formalis ratio</foreign> of quantity <lb/> +doth spring. +</s> +</p> +<p> +<s xml:id="echoid-s1020" xml:space="preserve"> +A finite space may be moved in infinite time. +</s> +<lb/> +<s xml:id="echoid-s1021" xml:space="preserve"> +There is a [¿]conditioned[?] +motion that a finite space <emph style="super">given</emph> <lb/> +cannot be moved <emph style="super">in a finite time</emph> but in an infinite time. +</s> +<lb/> +<s xml:id="echoid-s1022" xml:space="preserve"> +Also: that a finite space given cannot be moved +<emph style="super">in a finite time nor</emph> <lb/> +in an infinite time. +</s> +<lb/> +<s xml:id="echoid-s1023" xml:space="preserve"> +Also: that an infinite space may be moved <lb/> +in a finite time. +</s> +<lb/> +<s xml:id="echoid-s1024" xml:space="preserve"> +Also: that an infinite space <emph style="super">given</emph> may be moved not in <lb/> +a finite time but in an infinite time. +</s> +<lb/> +<s xml:id="echoid-s1025" xml:space="preserve"> +Also: that an infinite space given, may not be <lb/> +moved either in an infinite time nor finite. +</s> +</p> +<p> +<s xml:id="echoid-s1026" xml:space="preserve"> +Of contradictions that spring from diverse suppositions <lb/> +it cannot truly <emph style="st">[???]</emph> +<emph style="super">be</emph> sayd that the one parte +<emph style="st">doth [???]</emph> or <lb/> +other is false, for they are true consequently from <lb/> +there suppositions & in that respect are both true. but <lb/> +that which followeth is, that one of the suppositions <lb/> +is necessarily false, from where one of the <lb/> +partes of the contradiction was inferred. +</s> +</p> +<p> +<s xml:id="echoid-s1027" xml:space="preserve"> +As in the reason Achilles & other <lb/> +reasons of Zeno &c. +</s> +</p> +<pb file="add_6782_f374v" o="374v" n="749"/> +<pb file="add_6782_f375" o="375" n="750"/> +<pb file="add_6782_f375v" o="375v" n="751"/> +<pb file="add_6782_f376" o="376" n="752"/> +<pb file="add_6782_f376v" o="376v" n="753"/> +<pb file="add_6782_f377" o="377" n="754"/> +<pb file="add_6782_f377v" o="377v" n="755"/> +<pb file="add_6782_f378" o="378" n="756"/> +<pb file="add_6782_f378v" o="378v" n="757"/> +<pb file="add_6782_f379" o="379" n="758"/> +<pb file="add_6782_f379v" o="379v" n="759"/> +<pb file="add_6782_f380" o="380" n="760"/> +<head xml:id="echoid-head143" xml:space="preserve" xml:lang="lat"> +Proponatur, per Artem Analyticam solvere et inde <lb/> +componere, hoc problema: +<lb/>[<emph style="it">tr: +It is proposed, by the analytic are to solve and thence compose this problem. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1028" xml:space="preserve"> +<emph style="ul">Datam rectam terminatum: extrema ac media ratione secare.</emph> +<lb/>[<emph style="it">tr: +Given a finite straight line, to cut it in extreme and mean ratio. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1029" xml:space="preserve"> +Sit data recta terminata <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. et <lb/> +ponatur secari in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> ita ut <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math> sit minor pars, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> maior. +<lb/>[<emph style="it">tr: +Let the given finite straight line be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> and let it be cut in the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> +so that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math> is the lesser part and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> the greater. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1030" xml:space="preserve"> +Quoniam utraque pars ignota est, duæ possunt esse zeteses; etsi una <lb/> +sufficiat ad solutionem problematis. +<lb/>[<emph style="it">tr: +Since both parts are unkown, there are two possible zeteses, but one suffices for the solution of the problem. +</emph>]<lb/> +</s> +</p> +<head xml:id="echoid-head144" xml:space="preserve" xml:lang="lat"> +Zetesis. 1. +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1031" xml:space="preserve"> +ponatur primo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math> esse notum. +<lb/>[<emph style="it">tr: +first suppose <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math> is known. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1032" xml:space="preserve"> +Tum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>a</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>a</mi></mstyle></math>. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1033" xml:space="preserve"> +Tres igitur proportionales erunt: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore three proportionals are: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1034" xml:space="preserve"> +Inde resoluta analogia <lb/> +æaquatio erit. +<lb/>[<emph style="it">tr: +Whence having resolved the proportion, the equation will be: +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1035" xml:space="preserve"> +Et: per Antithesin: +<lb/>[<emph style="it">tr: +And by antithesis: +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1036" xml:space="preserve"> +Inde: Analogia. +<lb/>[<emph style="it">tr: +Whence, the ratio. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1037" xml:space="preserve"> +Ubi datur media proportionalis et adgregatum exremarum, <emph style="st">per</emph> <lb/> +ad Exegesin: +<lb/>[<emph style="it">tr: +Where there is given the mean proportional and the sum of the extremes, the resolution: +</emph>]<lb/> +</s> +</p> +<head xml:id="echoid-head145" xml:space="preserve" xml:lang="lat"> +Zetesis. 2. +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1038" xml:space="preserve"> +ponatur secundo, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> esse notam. +<lb/>[<emph style="it">tr: +suppose the second, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>, is known. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1039" xml:space="preserve"> +Tum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math> erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>-</mo><mi>a</mi><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math> will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>-</mo><mi>a</mi><mi>c</mi></mstyle></math>. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1040" xml:space="preserve"> +Tres igitur proportionales erunt. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>-</mo><mi>a</mi><mi>c</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> +<lb/>[<emph style="it">tr: +Therefore three proportionals will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>-</mo><mi>a</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1041" xml:space="preserve"> +Inde: resoluta analogia: erit. +<lb/>[<emph style="it">tr: +Whence the resolution of the ratio will be: +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1042" xml:space="preserve"> +Et: per Antithesin. +<lb/>[<emph style="it">tr: +And by antihesis: +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1043" xml:space="preserve"> +Inde: analogia: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Whence the ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi></mstyle></math> : <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> : <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> : <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1044" xml:space="preserve"> +Ubi datur media proportionalis et differentia extremarum, <lb/> +ad Exegesin. +<lb/>[<emph style="it">tr: +Where there is given a mean proportional and the difference of the extremes, the resolution. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f380v" o="380v" n="761"/> +<pb file="add_6782_f381" o="381" n="762"/> +<pb file="add_6782_f381v" o="381v" n="763"/> +<pb file="add_6782_f382" o="382" n="764"/> +<pb file="add_6782_f382v" o="382v" n="765"/> +<pb file="add_6782_f383" o="383" n="766"/> +<p> +<s xml:id="echoid-s1045" xml:space="preserve"> +to devide which by <lb/> +[???] a mean <lb/> +proportional. <lb/> +[???] <lb/> +the first being given is <lb/> +the same [???] all 3 to [???] <lb/> +the proportionals. +</s> +</p> +<pb file="add_6782_f383v" o="383v" n="767"/> +<pb file="add_6782_f384" o="384" n="768"/> +<pb file="add_6782_f384v" o="384v" n="769"/> +<pb file="add_6782_f385" o="385" n="770"/> +<pb file="add_6782_f385v" o="385v" n="771"/> +<pb file="add_6782_f386" o="386" n="772"/> +<pb file="add_6782_f386v" o="386v" n="773"/> +<pb file="add_6782_f387" o="387" n="774"/> +<pb file="add_6782_f387v" o="387v" n="775"/> +<pb file="add_6782_f388" o="388" n="776"/> +<div xml:id="echoid-div236" type="page_commentary" level="2" n="236"> +<p> +<s xml:id="echoid-s1046" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1046" xml:space="preserve"> +For a general explanation of the method see Add MS 6782, f. 399. <lb/> +The example on this page is a quartic equation with a fourth power, a linear term and a square, +the equation from Problem 6 of Viète's +<emph style="it">De numerosa potestatum resolutione</emph>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head146" xml:space="preserve" xml:lang="lat"> +12.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s1048" xml:space="preserve"> +prob. 6. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>d</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>f</mi><mi>f</mi><mi>f</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Problem 6. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>d</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>f</mi><mi>f</mi><mi>f</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1049" xml:space="preserve"> +Unicum <lb/> +Vietæ exemplum. +<lb/>[<emph style="it">tr: +Viète's only example. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1050" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>2</mn><mn>0</mn><mn>0</mn><mo>,</mo><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>0</mn><mn>0</mn><mi>a</mi><mo>=</mo><mn>4</mn><mn>4</mn><mn>9</mn><mn>3</mn><mn>7</mn><mn>6</mn></mstyle></math> +</s> +<lb/> +<s xml:id="echoid-s1051" xml:space="preserve"> +Species <lb/> +canonica. +<lb/>[<emph style="it">tr: +Canonical form. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1052" xml:space="preserve"> +Resolutio. +<lb/>[<emph style="it">tr: +Solution. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f388v" o="388v" n="777"/> +<pb file="add_6782_f389" o="389" n="778"/> +<div xml:id="echoid-div237" type="page_commentary" level="2" n="237"> +<p> +<s xml:id="echoid-s1053" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1053" xml:space="preserve"> +For a general explanation of the method see Add MS 6782, f. 399. <lb/> +The example on this page is a quartic equation with only a fourth power and a cube term, +the equation from Problem 5 of Viète's +<emph style="it">De numerosa potestatum resolutione</emph>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head147" xml:space="preserve" xml:lang="lat"> +11.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s1055" xml:space="preserve"> +prob. 5. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Problem 5. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1056" xml:space="preserve"> +Unicum <lb/> +Vietæ exemplum. +<lb/>[<emph style="it">tr: +Viète's only exmaple. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1057" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>0</mn><mo>,</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>0</mn><mn>0</mn><mi>a</mi><mo>=</mo><mn>4</mn><mn>7</mn><mn>0</mn><mn>0</mn><mn>1</mn><mn>6</mn></mstyle></math> +</s> +<lb/> +<s xml:id="echoid-s1058" xml:space="preserve"> +Species <lb/> +canonica. +<lb/>[<emph style="it">tr: +Canonical form. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1059" xml:space="preserve"> +Resolutio. +<lb/>[<emph style="it">tr: +solution. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1060" xml:space="preserve"> +Alterum exemplum nostrum: <lb/> +quod per divisionem. +<lb/>[<emph style="it">tr: +Another example of my own, done by division. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1061" xml:space="preserve"> +Resolutio. +<lb/>[<emph style="it">tr: +Solution. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1062" xml:space="preserve"> +pro 1<emph style="super">a</emph> figura <lb/> +Divide 44 <lb/> +per. 3 <lb/> +quotiens. 14. <lb/> +cuius maximus cubus 8 <lb/> +latus 2. <lb/> +pro 1<emph style="super">a</emph> figura <lb/> +si caetera <lb/> +consentiunt +<lb/>[<emph style="it">tr: +For the first figure, divide 44 by 3; the quotient is 14, whose greatest cube is 8, with side 2, +[to be taken] for the first figure, if the rest agree. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f389v" o="389v" n="779"/> +<pb file="add_6782_f390" o="390" n="780"/> +<div xml:id="echoid-div238" type="page_commentary" level="2" n="238"> +<p> +<s xml:id="echoid-s1063" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1063" xml:space="preserve"> +For a general explanation of the method see Add MS 6782, f. 399. <lb/> +The example on this page is a quartic equation with only a fourth power and a linear term, +the second equation from Problem 4 of Viète's +<emph style="it">De numerosa potestatum resolutione</emph>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head148" xml:space="preserve" xml:lang="lat"> +10.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s1065" xml:space="preserve"> +prob. 4. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Problem 4. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1066" xml:space="preserve"> +2. <lb/> +Vietæ exemplum. <lb/> +quod Per divisionem. +<lb/>[<emph style="it">tr: +Viète's second example, done by division. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1067" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>0</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>7</mn><mn>3</mn><mn>1</mn><mo>,</mo><mn>7</mn><mn>7</mn><mn>6</mn></mstyle></math> +</s> +<lb/> +<s xml:id="echoid-s1068" xml:space="preserve"> +Species <lb/> +canonica. +<lb/>[<emph style="it">tr: +Canonical form. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1069" xml:space="preserve"> +Resolutio. +<lb/>[<emph style="it">tr: +Solution. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f390v" o="390v" n="781"/> +<pb file="add_6782_f391" o="391" n="782"/> +<div xml:id="echoid-div239" type="page_commentary" level="2" n="239"> +<p> +<s xml:id="echoid-s1070" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1070" xml:space="preserve"> +For a general explanation of the method see Add MS 6782, f. 399. <lb/> +The example on this page is a quartic equation with only a fourth power and a linear term, +the first equation from Problem 4 of Viète's +<emph style="it">De numerosa potestatum resolutione</emph>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head149" xml:space="preserve" xml:lang="lat"> +9.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s1072" xml:space="preserve"> +prob. 4. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Problem 4. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi><mi>z</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1073" xml:space="preserve"> +1. <lb/> +Vietæ exemplum. +<lb/>[<emph style="it">tr: +Viète's first example. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1074" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>3</mn><mn>5</mn><mn>5</mn><mo>,</mo><mn>7</mn><mn>7</mn><mn>6</mn></mstyle></math> +</s> +<lb/> +<s xml:id="echoid-s1075" xml:space="preserve"> +Species <lb/> +canonica. +<lb/>[<emph style="it">tr: +Canonical form. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1076" xml:space="preserve"> +Resolutio. +<lb/>[<emph style="it">tr: +Solution. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f391v" o="391v" n="783"/> +<pb file="add_6782_f392" o="392" n="784"/> +<div xml:id="echoid-div240" type="page_commentary" level="2" n="240"> +<p> +<s xml:id="echoid-s1077" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1077" xml:space="preserve"> +For a general explanation of the method see Add MS 6782, f. 399. <lb/> +The example on this page is a cubic equation without a linear term, +the second equation from Problem 3 of Viète's +<emph style="it">De numerosa potestatum resolutione</emph>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head150" xml:space="preserve" xml:lang="lat"> +8.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s1079" xml:space="preserve"> +prob. 3. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Problem 3. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1080" xml:space="preserve"> +2. <lb/> +Vietæ exemplum <lb/> +quod per divisionem. +<lb/>[<emph style="it">tr: +Viète's second example, done by division. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1081" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>1</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>,</mo><mi>a</mi><mi>a</mi><mo>=</mo><mn>5</mn><mo>,</mo><mn>7</mn><mn>7</mn><mn>3</mn><mo>,</mo><mn>8</mn><mn>2</mn><mn>4</mn></mstyle></math> +</s> +<lb/> +<s xml:id="echoid-s1082" xml:space="preserve"> +Species <lb/> +canonica. +<lb/>[<emph style="it">tr: +Canonical form. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1083" xml:space="preserve"> +Resolutio. +<lb/>[<emph style="it">tr: +Solution. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1084" xml:space="preserve"> +Nota. +<lb/>[<emph style="it">tr: +Note. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1085" xml:space="preserve"> +prima figura acquiritur <lb/> +per divisionem ac si <lb/> +Species canonica <lb/> +esset: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>d</mi><mo>+</mo><mi>b</mi><mo maxsize="1">)</mo><mo>×</mo><mi>b</mi><mi>b</mi></mstyle></math> quæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>b</mi><mi>b</mi><mi>d</mi><mo>+</mo><mi>b</mi><mi>b</mi><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +the first figure is acquired by division as if the canonical form were <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>d</mi><mo>+</mo><mi>b</mi><mo maxsize="1">)</mo><mo>×</mo><mi>b</mi><mi>b</mi></mstyle></math> quæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>b</mi><mi>b</mi><mi>d</mi><mo>+</mo><mi>b</mi><mi>b</mi><mi>b</mi></mstyle></math>. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1086" xml:space="preserve"> +hoc est si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>+</mo><mi>b</mi></mstyle></math> sit <lb/> +divisor <lb/> +Quotiens erit, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math> +<lb/>[<emph style="it">tr: +that is, if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>+</mo><mi>b</mi></mstyle></math> is the divisor, the quotient will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math>. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1087" xml:space="preserve"> +Et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> latus erit 1<emph style="super">a</emph> figura +<lb/>[<emph style="it">tr: +And the square-root <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> will be the first figure. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1088" xml:space="preserve"> +Ut hic Quotiens est 5. <lb/> +cuius latus <emph style="super">quadratum</emph> 2. pro <lb/> +prima figura. +<lb/>[<emph style="it">tr: +As here the quotient is 5, whose square-root is 2 for the first figure. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1089" xml:space="preserve"> +Vide Vietam. +<lb/>[<emph style="it">tr: +See Viète. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f392v" o="392v" n="785"/> +<pb file="add_6782_f393" o="393" n="786"/> +<div xml:id="echoid-div241" type="page_commentary" level="2" n="241"> +<p> +<s xml:id="echoid-s1090" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1090" xml:space="preserve"> +For a general explanation of the method see Add MS 6782, f. 399. <lb/> +The example on this page is a cubic equation without a linear term, +the first equation from Problem 3 of Viète's +<emph style="it">De numerosa potestatum resolutione</emph>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head151" xml:space="preserve" xml:lang="lat"> +7.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s1092" xml:space="preserve"> +prob. 3. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Problem 3. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1093" xml:space="preserve"> +1. <lb/> +Vietæ exemplum +<lb/>[<emph style="it">tr: +Viète's first example +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1094" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>3</mn><mn>0</mn><mo>,</mo><mi>a</mi><mi>a</mi><mo>=</mo><mn>8</mn><mn>6</mn><mo>,</mo><mn>2</mn><mn>2</mn><mn>0</mn><mo>,</mo><mn>2</mn><mn>8</mn><mn>8</mn></mstyle></math>. +</s> +<lb/> +<s xml:id="echoid-s1095" xml:space="preserve"> +Species <lb/> +canonica. +<lb/>[<emph style="it">tr: +Canonical form. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1096" xml:space="preserve"> +Resolutio. +<lb/>[<emph style="it">tr: +Solution. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1097" xml:space="preserve"> +Jam 43. fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Now <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> becomes 43. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f393v" o="393v" n="787"/> +<pb file="add_6782_f394" o="394" n="788"/> +<div xml:id="echoid-div242" type="page_commentary" level="2" n="242"> +<p> +<s xml:id="echoid-s1098" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1098" xml:space="preserve"> +For a general explanation of the method see Add MS 6782, f. 399. <lb/> +The example on this page is a cubic equation without a square term, +the second equation from Problem 2 of Viète's +<emph style="it">De numerosa potestatum resolutione</emph>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head152" xml:space="preserve" xml:lang="lat"> +6.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s1100" xml:space="preserve"> +prob. 2. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Problem 2. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. +</emph>]<lb/></s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1101" xml:space="preserve"> +2. <lb/> +Vietæ exemplum +<lb/>[<emph style="it">tr: +Viète's second example. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1102" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>9</mn><mn>5</mn><mn>4</mn><mn>0</mn><mn>0</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>8</mn><mn>1</mn><mn>9</mn><mo>,</mo><mn>4</mn><mn>5</mn><mn>9</mn></mstyle></math> +</s> +<lb/> +<s xml:id="echoid-s1103" xml:space="preserve"> +Species <lb/> +canonica +<lb/>[<emph style="it">tr: +Canonical form +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1104" xml:space="preserve"> +Resolutio: +<lb/>[<emph style="it">tr: +Solution. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1105" xml:space="preserve"> +per divisionem. +<lb/>[<emph style="it">tr: +By division. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f394v" o="394v" n="789"/> +<pb file="add_6782_f395" o="395" n="790"/> +<div xml:id="echoid-div243" type="page_commentary" level="2" n="243"> +<p> +<s xml:id="echoid-s1106" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1106" xml:space="preserve"> +For a general explanation of the method see Add MS 6782, f. 399. <lb/> +The example on this page is a cubic equation without a square term, +the first equation from Problem 2 of Viète's +<emph style="it">De numerosa potestatum resolutione</emph>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head153" xml:space="preserve" xml:lang="lat"> +5.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s1108" xml:space="preserve"> +prob. 2. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Problem 2. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1109" xml:space="preserve"> +1. <lb/> +Vietæ exemplum +<lb/>[<emph style="it">tr: +Viète's first example. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1110" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>3</mn><mn>0</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>1</mn><mn>4</mn><mn>3</mn><mn>5</mn><mn>6</mn><mn>1</mn><mn>9</mn><mn>7</mn></mstyle></math>. +</s> +<lb/> +<s xml:id="echoid-s1111" xml:space="preserve"> +Species <lb/> +canonica. +<lb/>[<emph style="it">tr: +Canonical form. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1112" xml:space="preserve"> +Vel: +<lb/>[<emph style="it">tr: +Or: +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1113" xml:space="preserve"> +Resolutio. +<lb/>[<emph style="it">tr: +Solution. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1114" xml:space="preserve"> +Vide supra. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +See above at A. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1115" xml:space="preserve"> +Jam 24 fiat. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Now <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> becomes 24. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f395v" o="395v" n="791"/> +<pb file="add_6782_f396" o="396" n="792"/> +<div xml:id="echoid-div244" type="page_commentary" level="2" n="244"> +<p> +<s xml:id="echoid-s1116" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1116" xml:space="preserve"> +For a general explanation of the method see Add MS 6782, f. 399. <lb/> +Here the method is applied to a cubic equation without a square term. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head154" xml:space="preserve" xml:lang="lat"> +4.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s1118" xml:space="preserve"> +prob. 2. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Problem 2. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1119" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>3</mn><mn>5</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>2</mn><mn>2</mn><mn>9</mn><mn>3</mn><mn>2</mn></mstyle></math> +</s> +<lb/> +<s xml:id="echoid-s1120" xml:space="preserve"> +Species <lb/> +canonica +<lb/>[<emph style="it">tr: +Canonical form +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1121" xml:space="preserve"> +Hoc est: +<lb/>[<emph style="it">tr: +That is: +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1122" xml:space="preserve"> +Vel. +<lb/>[<emph style="it">tr: +Or. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1123" xml:space="preserve"> +Resolutio. +<lb/>[<emph style="it">tr: +Solution. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f396v" o="396v" n="793"/> +<pb file="add_6782_f397" o="397" n="794"/> +<div xml:id="echoid-div245" type="page_commentary" level="2" n="245"> +<p> +<s xml:id="echoid-s1124" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1124" xml:space="preserve"> +For a general explanation of the method see Add MS 6782, f. 399. <lb/> +In the example on this page, the large size of the coefficient in relation to the root means that +the term <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mn>6</mn><mn>2</mn><mi>b</mi></mstyle></math> must be taken into account in determining the first digit. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head155" xml:space="preserve" xml:lang="lat"> +3.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1126" xml:space="preserve"> +prob. 1. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>z</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Problem 1. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>z</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1127" xml:space="preserve"> +Casus 2<emph style="super">a</emph>. <lb/> +per divisionem +<lb/>[<emph style="it">tr: +Case 2, by division +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1128" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mn>7</mn><mn>6</mn><mn>2</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>2</mn><mn>2</mn><mn>1</mn><mn>2</mn><mn>0</mn></mstyle></math> +</s> +<lb/> +<s xml:id="echoid-s1129" xml:space="preserve"> +Species <lb/> +canonica <lb/> +ut supra. +<lb/>[<emph style="it">tr: +Canonical form as above. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1130" xml:space="preserve"> +Vel: +<lb/>[<emph style="it">tr: +Or: +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1131" xml:space="preserve"> +Resolutio. +<lb/>[<emph style="it">tr: +Solution +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1132" xml:space="preserve"> +Quoniam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mo>></mo><mn>2</mn></mstyle></math> <lb/> +fiat devolutio. +<lb/>[<emph style="it">tr: +Because <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>7</mn><mo>></mo><mn>2</mn></mstyle></math> it becomes a devolution. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1133" xml:space="preserve"> +Vietæ exemplum. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mn>9</mn><mn>5</mn><mn>4</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>1</mn><mn>8</mn><mn>4</mn><mn>8</mn><mn>7</mn></mstyle></math> +<lb/>[<emph style="it">tr: +Viète's example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mn>9</mn><mn>5</mn><mn>4</mn><mi>a</mi><mo>=</mo><mn>1</mn><mn>8</mn><mn>4</mn><mn>8</mn><mn>7</mn></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f397v" o="397v" n="795"/> +<pb file="add_6782_f398" o="398" n="796"/> +<div xml:id="echoid-div246" type="page_commentary" level="2" n="246"> +<p> +<s xml:id="echoid-s1134" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1134" xml:space="preserve"> +For a general explanation of the method see Add MS 6782, f. 399. <lb/> +The equation on this page has a 3-digit solution. The first digit is found by inspection to be 2 (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mn>2</mn><mn>0</mn><mn>0</mn></mstyle></math>). +The second digit, is found to be 4 (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>4</mn><mn>0</mn></mstyle></math>). +The process is now repeated, treating <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> as a single quantity, to find the third digit, again labelled <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +This is found to be 3. +For Harriot's owon description of this process see Add MS 6784, f. 408. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head156" xml:space="preserve" xml:lang="lat"> +2.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1136" xml:space="preserve"> +prob. 1. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>z</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Problem 1. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>z</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1137" xml:space="preserve"> +Exemplum primum <lb/> +Vietæ. +<lb/>[<emph style="it">tr: +Viète's first example +</emph>]<lb/> +[<emph style="it">Note: +This is Problem 1 from Viète's <emph style="it">De numerosa potestatum resolutione</emph>. + </emph>]<lb/> +</s> +</p> +<p> +<s xml:id="echoid-s1138" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mn>7</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>6</mn><mn>0</mn><mn>7</mn><mn>5</mn><mn>0</mn></mstyle></math> +</s> +<lb/> +<s xml:id="echoid-s1139" xml:space="preserve"> +Species <lb/> +canonica. <lb/> +ut supra. +<lb/>[<emph style="it">tr: +Canonical form as above. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1140" xml:space="preserve"> +hoc est: +<lb/>[<emph style="it">tr: +that is: +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1141" xml:space="preserve"> +Resolutio Vietana. paucis mutatis +<lb/>[<emph style="it">tr: +Viète's solution, a little changed +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f398v" o="398v" n="797"/> +<pb file="add_6782_f399" o="399" n="798"/> +<div xml:id="echoid-div247" type="page_commentary" level="2" n="247"> +<p> +<s xml:id="echoid-s1142" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1142" xml:space="preserve"> +This is the first of a set of 12 pages on extracting roots of positively affected equations, +that is, equations where all the powers following the first are positive. +Such equations have one, and only one, positive root. <lb/> +The work is closely based on Problems 1 to 6 in +Viète, <emph style="it">De numerosa potestatum ad exegesin resolutione</emph> (1600); +Harriot's heading 'De numerosa potestatum resolutione' directly echoes Viète's. <lb/> +The method works by finding each digit of the root in turn. +Suppose that a required root <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> of is of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>, +where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> represents multiples of 10 and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> represents units. +The first digit is found by inspection. The canonical form then shows how to estimate <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. <lb/> +In the problem on this page, for example, Harriot first takes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> to be 40, but quickly finds that this is too large. +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> must be 30. Subtracting the known values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math> (= 900) and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi></mstyle></math> (= 720) from 2356 +leaves him with 736, which according to the canonical form must correspond to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>d</mi><mo>+</mo><mi>c</mi><mi>c</mi></mstyle></math>. +A first estimate for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is found by dividing 736 by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>b</mi><mo>+</mo><mi>d</mi></mstyle></math> (= 84). The integer part of the quotient is 8. +In fact <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>8</mn></mstyle></math> satisfies the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>d</mi><mo>+</mo><mi>c</mi><mi>c</mi><mo>=</mo><mn>7</mn><mn>3</mn><mn>6</mn></mstyle></math> exactly and so the process is complete. <lb/> +Later examples become more complicated but follow the same basic procedure. <lb/> +For further discussion see See Stedall 2003, 45–62 and 292, and Stedall 2011, 29–31. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head157" xml:space="preserve" xml:lang="lat"> +1.) De numerosa potestatum resolutione. Vieta. fol. 7. b. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. Viète, folio 7b. +</emph>]<lb/> +</head> +<p> +<s xml:id="echoid-s1144" xml:space="preserve"> +prob. 1. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>z</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Problem 1. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mi>d</mi><mi>a</mi><mo>=</mo><mi>x</mi><mi>z</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1145" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mn>2</mn><mn>4</mn><mo>,</mo><mi>a</mi><mo>=</mo><mn>2</mn><mn>3</mn><mn>5</mn><mn>6</mn></mstyle></math>. +</s> +<lb/> +<s xml:id="echoid-s1146" xml:space="preserve"> +Species <lb/> +canonica +<lb/>[<emph style="it">tr: +Canonical form +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1147" xml:space="preserve"> +Hoc est: +<lb/>[<emph style="it">tr: +That is: +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1148" xml:space="preserve"> +vel: +<lb/>[<emph style="it">tr: +or: +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1149" xml:space="preserve"> +Genesis Vulgaris +<lb/>[<emph style="it">tr: +Common derivation +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1150" xml:space="preserve"> +Genesis Specialis <lb/> +seu canonica +<lb/>[<emph style="it">tr: +Specific or canonical derivation. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1151" xml:space="preserve"> +Resolutio secundum methodum Vietanam, <lb/> +paucis mutatis +<lb/>[<emph style="it">tr: +Solution according to the method of Viète, a little changed +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1152" xml:space="preserve"> +non potest auferri. <lb/> +operatio igitur iteranda <lb/> +et fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. minor. videlicet. 3. +<lb/>[<emph style="it">tr: +cannot be subtracted; the work must therefore be repeated making <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> smaller, namely, 3. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1153" xml:space="preserve"> +Aliter. +<lb/>[<emph style="it">tr: +Another way. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f399v" o="399v" n="799"/> +<pb file="add_6782_f400" o="400" n="800"/> +<div xml:id="echoid-div248" type="page_commentary" level="2" n="248"> +<p> +<s xml:id="echoid-s1154" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1154" xml:space="preserve"> +Following from the general treatment in Add MS 6782, f. 403, f. 402, f. 401, +of avulsed quartics with no square or linear term, +Harriot here solves the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>4</mn><mn>8</mn><mn>1</mn><mn>5</mn><mn>4</mn><mn>4</mn><mo>=</mo><mn>6</mn><mn>5</mn><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> for both roots (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn><mn>8</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>5</mn><mn>7</mn></mstyle></math>). +He also shows how either root may be obtained from the other. <lb/> +The equation is taken from Problem 20 of Viète, +<emph style="it">De numerosa potestatum ad exegesin resolutione</emph>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head158" xml:space="preserve" xml:lang="lat"> +c.17.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1156" xml:space="preserve"> +prob. 20. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex. +<lb/>[<emph style="it">tr: +Problem 20. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1157" xml:space="preserve"> +canon ad <lb/> +resolutionem. +<lb/>[<emph style="it">tr: +Canonical form for the solution. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1158" xml:space="preserve"> +Resolutio. +<lb/>[<emph style="it">tr: +Solution. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1159" xml:space="preserve"> +Eductio radicis <lb/> +minoris. +<lb/>[<emph style="it">tr: +Extraction of the smaller root. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1160" xml:space="preserve"> +Radix igitur <lb/> +minor. 38. +<lb/>[<emph style="it">tr: +Therefore the smaller root is 38. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1161" xml:space="preserve"> +Quæratur maior. +<lb/>[<emph style="it">tr: +The larger root is sought. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1162" xml:space="preserve"> +Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, minor <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> maior. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the smaller root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> the larger. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1163" xml:space="preserve"> +Vieta aliter <lb/> +ut pag: supra. +<lb/>[<emph style="it">tr: +Viète otherwise, as in the page above. +</emph>]<lb/> +[<emph style="it">Note: +The page above is Add MS 6782, f. 402. + </emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1164" xml:space="preserve"> +Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. 57. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>5</mn><mn>7</mn></mstyle></math> is given. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1165" xml:space="preserve"> +Eductio radicis <lb/> +maioris. +<lb/>[<emph style="it">tr: +Extraction of the larger root. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1166" xml:space="preserve"> +Radix igitur maior. 57. +<lb/>[<emph style="it">tr: +Therefore the larger root is 57. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1167" xml:space="preserve"> +Quæratur iam minor. +<lb/>[<emph style="it">tr: +Now the smaller root is sought. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1168" xml:space="preserve"> +Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> minor, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> minor. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> be the smaller root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> the larger. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1169" xml:space="preserve"> +Vieta aliter <lb/> +ut pag: supra. +<lb/>[<emph style="it">tr: +Viète otherwise as in the page above. +</emph>]<lb/> +[<emph style="it">Note: +The page above is Add MS 6782, f. 402. + </emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1170" xml:space="preserve"> +Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. 38. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mn>3</mn><mn>8</mn></mstyle></math> is given. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f400v" o="400v" n="801"/> +<pb file="add_6782_f401" o="401" n="802"/> +<div xml:id="echoid-div249" type="page_commentary" level="2" n="249"> +<p> +<s xml:id="echoid-s1171" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1171" xml:space="preserve"> +In this final page of Section c, Harriot argues that the first figure of the larger root must be 5. +Assuming that it is either 4 or 6 will lead to a contradiction. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head159" xml:space="preserve" xml:lang="lat"> +c.18.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1173" xml:space="preserve"> +prob. 20. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>,</mo><mn>4</mn><mn>8</mn><mn>1</mn><mo>,</mo><mn>5</mn><mn>4</mn><mn>4</mn><mo>=</mo><mn>6</mn><mn>5</mn><mo>,</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>.. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex. +<lb/>[<emph style="it">tr: +Problem 20. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>4</mn><mn>8</mn><mn>1</mn><mn>5</mn><mn>4</mn><mn>4</mn><mo>=</mo><mn>6</mn><mn>5</mn><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>.. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1174" xml:space="preserve"> +<emph style="st">Additementum <lb/> +nostrum</emph> <lb/> +Absurdum consequens in eductione lateris maioris <lb/> +si prima figura sit minor vel maior. 5. +<lb/>[<emph style="it">tr: +Nonsensical consequences in the extraction of the larger root if the first figure is less than or greater than 5. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1175" xml:space="preserve"> +B. Divisor <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mn>4</mn><mn>1</mn><mn>0</mn></mstyle></math> +<lb/>[<emph style="it">tr: +The divisor is 5410. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1176" xml:space="preserve"> +Absurdum. +<lb/>[<emph style="it">tr: +Contradiction +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1177" xml:space="preserve"> +Nam cum Residuum supra <lb/> +habent signum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo></mstyle></math> <lb/> +et Divisor <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo></mstyle></math> <lb/> +Parabola erit etiam, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo></mstyle></math>. +<lb/>[<emph style="it">tr: +For with the above residue we have a <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo></mstyle></math> sign, while the divisor is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo></mstyle></math>, +so the comparison is also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1178" xml:space="preserve"> +Prima figura igitur <lb/> +non est 4. +<lb/>[<emph style="it">tr: +Therefore the first figure is not 4. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1179" xml:space="preserve"> +B. Divisor <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mn>1</mn><mn>7</mn><mn>2</mn><mn>0</mn><mn>7</mn><mn>5</mn></mstyle></math> +<lb/>[<emph style="it">tr: +The divisor is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mn>1</mn><mn>7</mn><mn>2</mn><mn>0</mn><mn>7</mn><mn>5</mn></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1180" xml:space="preserve"> +Absurdum. +<lb/>[<emph style="it">tr: +Contradiction. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1181" xml:space="preserve"> +Nam cum residuum supra <lb/> +habent signum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo></mstyle></math> <lb/> +et divisor, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo></mstyle></math> <lb/> +parabola erit etiam, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo></mstyle></math>. +<lb/>[<emph style="it">tr: +For while the above residue has the sign <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo></mstyle></math> and the divisor <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo></mstyle></math>, the comparison is also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1182" xml:space="preserve"> +Prima figura igitur <lb/> +non est, 6. +<lb/>[<emph style="it">tr: +Therefore the first figure is not 6. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1183" xml:space="preserve"> +Erit igitur 5. per limitum <lb/> +præfinitiones, sicut <lb/> +per exemplum est explicatum. +<lb/>[<emph style="it">tr: +Therefore it will be 5, by the determinations of limits, as explained by the example. +</emph>]<lb/> +</s> +</p> +<head xml:id="echoid-head160" xml:space="preserve" xml:lang="lat"> +Nota. +<lb/>[<emph style="it">tr: +Note +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1184" xml:space="preserve"> +Operatio pro eductione lateris maioris aliquando secundum partem <lb/> +est similis operationem pro eductione lateris minoris. +<lb/>[<emph style="it">tr: +In the operation for extracting the larger root, +sometimes the second part is similar to the operation for extracting the smaller root. +</emph>]<lb/> +</s> +<s xml:id="echoid-s1185" xml:space="preserve"> +Nimirum si <lb/> +utraque latera consentiant in primus figurus +<emph style="super">et numero figurum</emph>. +Evidently, if both roots agree in the first figure and the number of figures. +</s> +<s xml:id="echoid-s1186" xml:space="preserve"> +Ut si latus minus <lb/> +sit 23. maius 24. +<lb/>[<emph style="it">tr: +As if the smaller root is 23, the larger is 24. +</emph>]<lb/> +</s> +<s xml:id="echoid-s1187" xml:space="preserve"> +Ita si minus 343, maius 347. est sic de cæteris. +<lb/>[<emph style="it">tr: +Thus if the smaller root is 343, the larger 347, and so on for others. +</emph>]<lb/> +</s> +<s xml:id="echoid-s1188" xml:space="preserve"> +Sed si dissentiant in primum figura: dissimiles erunt operationes totaliter <lb/> +ut in exemplis antecedentis, in his chartis expositis. +<lb/>[<emph style="it">tr: +But if they do not agree in the first figure, the operations will be completely different, +as in the preceding examples explained in these sheets. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f401v" o="401v" n="803"/> +<pb file="add_6782_f402" o="402" n="804"/> +<div xml:id="echoid-div250" type="page_commentary" level="2" n="250"> +<p> +<s xml:id="echoid-s1189" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1189" xml:space="preserve"> +On this page Harriot compares his own method with that of Viète, in Problem 20 of +<emph style="it">De numerosa potestatum ad exegesin resolutione</emph>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head161" xml:space="preserve" xml:lang="lat"> +c.16.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1191" xml:space="preserve"> +prob. 20. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex. +<lb/>[<emph style="it">tr: +Problem 20. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1192" xml:space="preserve"> +Species canonica <lb/> +ad radices inæquales. +<lb/>[<emph style="it">tr: +Canonical form for unequal roots. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1193" xml:space="preserve"> +Aliter quam supra, <lb/> +et ut Vieta. +<lb/>[<emph style="it">tr: +Another way from that above, and as Viète. +</emph>]<lb/> +[<emph style="it">Note: +By 'above' Harriot means the working given in Add MS 6782, f. 404, f. 403. + </emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1194" xml:space="preserve"> +Si una radix sit nota, <lb/> +altera erit cognita. +<lb/>[<emph style="it">tr: +If one roots is known, the other will be known. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1195" xml:space="preserve"> +4<emph style="super">or</emph> continue proportionales. +Four continued proportionals. +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1196" xml:space="preserve"> +Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> nota. Quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is sought. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1197" xml:space="preserve"> +Sint continue proportionales +<lb/>[<emph style="it">tr: +Let there be continued proportinals +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1198" xml:space="preserve"> +datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> et inde <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> is given and thence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1199" xml:space="preserve"> +erit: <lb/> +inde: +<lb/>[<emph style="it">tr: +We will have: <lb/> +thence: +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1200" xml:space="preserve"> +Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> nota. Quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is sought. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1201" xml:space="preserve"> +Sint continue proportionales +<lb/>[<emph style="it">tr: +Let there be continued proportinals +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1202" xml:space="preserve"> +datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> et inde, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> is given, and thence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1203" xml:space="preserve"> +erit: <lb/> +inde: +<lb/>[<emph style="it">tr: +We will have: <lb/> +thence: +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f402v" o="402v" n="805"/> +<pb file="add_6782_f403" o="403" n="806"/> +<div xml:id="echoid-div251" type="page_commentary" level="2" n="251"> +<p> +<s xml:id="echoid-s1204" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1204" xml:space="preserve"> +On this page, Harriot continues his general treatment of equations of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, +begun on the previous page. <lb/> +The numerical example at the end, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>4</mn><mn>8</mn><mn>1</mn><mn>5</mn><mn>4</mn><mn>4</mn><mo>=</mo><mn>6</mn><mn>5</mn><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, is from Problem 20 of Viète, +<emph style="it">De numerosa potestatum ad exegesin resolutione</emph>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head162" xml:space="preserve" xml:lang="lat"> +c.15.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1206" xml:space="preserve"> +prob. 20. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex. +<lb/>[<emph style="it">tr: +Problem 20. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1207" xml:space="preserve"> +Species canonica <lb/> +ad radices inæquales. +<lb/>[<emph style="it">tr: +Canonical form for unequal roots. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1208" xml:space="preserve"> +Si una radix sit nota, <lb/> +altera erit cognita. +<lb/>[<emph style="it">tr: +If one root is known, the other will be known. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1209" xml:space="preserve"> +sit, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> nota. Quæaeritur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is sought. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1210" xml:space="preserve"> +Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is given. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1211" xml:space="preserve"> +Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is given. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1212" xml:space="preserve"> +sit, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> nota. Quæaeritur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is sought. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1213" xml:space="preserve"> +Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is given. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1214" xml:space="preserve"> +Unde eadem æquatio: +<lb/>[<emph style="it">tr: +Whence the same equation: +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1215" xml:space="preserve"> +Datur <emph style="st">igitur</emph> igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is given. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1216" xml:space="preserve"> +Poristicum +<lb/>[<emph style="it">tr: +Proof +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1217" xml:space="preserve"> +Quod, +<lb/>[<emph style="it">tr: +Because +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1218" xml:space="preserve"> +Est enim: est igitur. +<lb/>[<emph style="it">tr: +Indeed it is. Therefore it is so. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1219" xml:space="preserve"> +Pro exemplo ad resolutionem. +<lb/>[<emph style="it">tr: +According to this example for the solution. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1220" xml:space="preserve"> +In numeris, sit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>.</mo><mn>3</mn><mn>8</mn></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>.</mo><mn>5</mn><mn>7</mn><mo>.</mo></mstyle></math> +<lb/>[<emph style="it">tr: +In numbers let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mn>3</mn><mn>8</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>5</mn><mn>7</mn></mstyle></math>. +</emph>]<lb/> +</s> +<s xml:id="echoid-s1221" xml:space="preserve"> +Hoc est: <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>,</mo><mn>4</mn><mn>8</mn><mn>1</mn><mo>,</mo><mn>5</mn><mn>4</mn><mn>4</mn><mo>=</mo><mn>6</mn><mn>5</mn><mo>,</mo><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +That is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>4</mn><mn>8</mn><mn>1</mn><mn>5</mn><mn>4</mn><mn>4</mn><mo>=</mo><mn>6</mn><mn>5</mn><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1222" xml:space="preserve"> +Limites radicum. +<lb/>[<emph style="it">tr: +The limits of the roots. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f403v" o="403v" n="807"/> +<pb file="add_6782_f404" o="404" n="808"/> +<div xml:id="echoid-div252" type="page_commentary" level="2" n="252"> +<p> +<s xml:id="echoid-s1223" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1223" xml:space="preserve"> +On this page Harriot begins a general treatment of equations of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>d</mi><mi>a</mi><mi>a</mi></mstyle></math>, +with no square or linear term. +In order to preserve dimensions, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math> is used as a placeholder for a general 4-dimensional quantity. +All coeffcients are assumed to be positive. +Equations of this kind have two positive roots or none at all, depending on the size of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. <lb/> +For Harriot's derivation of the canonical form for unequal roots, see Add MS 6783, f. 173. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head163" xml:space="preserve" xml:lang="lat"> +c.14.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1225" xml:space="preserve"> +prob. 20. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex. +<lb/>[<emph style="it">tr: +Problem 20. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1226" xml:space="preserve"> +Species canonica <lb/> +ad radices inæquales. +<lb/>[<emph style="it">tr: +Canonical form for unequal roots. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1227" xml:space="preserve"> +Nam: Si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>. erit: +<lb/>[<emph style="it">tr: +For if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math> then: +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1228" xml:space="preserve"> +Et ita est: +<lb/>[<emph style="it">tr: +And so it is. +</emph>]<lb/> +</s> +<s xml:id="echoid-s1229" xml:space="preserve"> +Si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math>. <lb/> +erit: +<lb/>[<emph style="it">tr: +If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math> then: +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1230" xml:space="preserve"> +est igitur +<lb/>[<emph style="it">tr: +Therefore it is so. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1231" xml:space="preserve"> +Species ad radices <lb/> +æquales. +<lb/>[<emph style="it">tr: +Canonical form for equal roots. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1232" xml:space="preserve"> +Sunt continue proportionalia <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math> +<lb/>[<emph style="it">tr: +There are continued proportionals <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1233" xml:space="preserve"> +Sit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, minor radix. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, maior. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the smaller root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> the larger. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1234" xml:space="preserve"> +fiat: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mo>,</mo><mn>3</mn><mo>:</mo><mi>d</mi><mo>,</mo><mfrac><mrow><mn>3</mn><mi>d</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> +<lb/>[<emph style="it">tr: +let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mo>:</mo><mn>3</mn><mo>=</mo><mi>d</mi><mo>:</mo><mfrac><mrow><mn>3</mn><mi>d</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1235" xml:space="preserve"> +Dico quod: +<lb/>[<emph style="it">tr: +I say that +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1236" xml:space="preserve"> +Est enim. est igitur. +<lb/>[<emph style="it">tr: +Indeed it is. Therefore it is so. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1237" xml:space="preserve"> +Dico quod: <lb/> +vel: +<lb/>[<emph style="it">tr: +I say that: <lb/> +or: +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1238" xml:space="preserve"> +Est enim. est igitur. +<lb/>[<emph style="it">tr: +Indeed it is. Therefore it is so. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1239" xml:space="preserve"> +Dico quod: <lb/> +vel: +<lb/>[<emph style="it">tr: +I say that: <lb/> +or: +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1240" xml:space="preserve"> +Est enim. <lb/> +Est igitur. +<lb/>[<emph style="it">tr: +Indeed it is. Therefore it is so. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1241" xml:space="preserve"> +ergo. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>></mo><mi>a</mi></mstyle></math>. Hoc est qualibet <lb/> +radice. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>></mo><mi>a</mi></mstyle></math>. This is so whatever the root. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f404v" o="404v" n="809"/> +<pb file="add_6782_f405" o="405" n="810"/> +<div xml:id="echoid-div253" type="page_commentary" level="2" n="253"> +<p> +<s xml:id="echoid-s1242" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1242" xml:space="preserve"> +Following from the general treatment in Add MS 6782, f. 407, f. 406, +of avulsed quartics with no cube or square term, +Harriot here solves the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>1</mn><mn>7</mn><mn>9</mn><mn>4</mn><mn>4</mn><mo>=</mo><mn>2</mn><mn>7</mn><mn>7</mn><mn>5</mn><mn>5</mn><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> for both roots (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>8</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn><mn>7</mn></mstyle></math>). +He also shows how either root may be obtained from the other. <lb/> +The equation is taken from Problem 19 of Viète, +<emph style="it">De numerosa potestatum ad exegesin resolutione</emph>. +Viète gave rules for the relationship between the two roots but did not explain how he had arrived at them. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head164" xml:space="preserve" xml:lang="lat"> +c.13.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1244" xml:space="preserve"> +prob. 19. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex. +<lb/>[<emph style="it">tr: +Problem 19. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1245" xml:space="preserve"> +Canon ad <lb/> +resolutionem +<lb/>[<emph style="it">tr: +Canonical form for the solution. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1246" xml:space="preserve"> +Resolutio. +<lb/>[<emph style="it">tr: +Solution +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1247" xml:space="preserve"> +Eductio radicis <lb/> +Minoris. +<lb/>[<emph style="it">tr: +Extraction of the smaller root. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1248" xml:space="preserve"> +Radix igitur minor est 8. +<lb/>[<emph style="it">tr: +Therefore the smaller root is 8. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1249" xml:space="preserve"> +Quæratur iam maior. +<lb/>[<emph style="it">tr: +Now the larger root is sought. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1250" xml:space="preserve"> +Sit minor <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. maior <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the smaller root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> the larger. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1251" xml:space="preserve"> +Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. 27. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>2</mn><mn>7</mn></mstyle></math> is given. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1252" xml:space="preserve"> +Eductio radicis <lb/> +Maioris. +<lb/>[<emph style="it">tr: +Extraction of the larger root. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f405v" o="405v" n="811"/> +<pb file="add_6782_f406" o="406" n="812"/> +<div xml:id="echoid-div254" type="page_commentary" level="2" n="254"> +<p> +<s xml:id="echoid-s1253" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1253" xml:space="preserve"> +On this page, Harriot continues his general treatment of equations of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, +begun on the previous page. <lb/> +The numerical example at the end, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>1</mn><mn>7</mn><mn>9</mn><mn>4</mn><mn>4</mn><mo>=</mo><mn>2</mn><mn>7</mn><mn>7</mn><mn>5</mn><mn>5</mn><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, is from Problem 19 of Viète, +<emph style="it">De numerosa potestatum ad exegesin resolutione</emph>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head165" xml:space="preserve" xml:lang="lat"> +c.12.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1255" xml:space="preserve"> +prob. 19. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex. +<lb/>[<emph style="it">tr: +Problem 19. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1256" xml:space="preserve"> +Species canonica <lb/> +ad radices inæquales. +<lb/>[<emph style="it">tr: +Canonical form for unequal roots. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1257" xml:space="preserve"> +Si una radix sit nota, <lb/> +altera erit cognita. +<lb/>[<emph style="it">tr: +If one root is known, the other will be known. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1258" xml:space="preserve"> +Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> nota. Quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is sought. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1259" xml:space="preserve"> +Datur igitur, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is given. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1260" xml:space="preserve"> +Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is given. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1261" xml:space="preserve"> +Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. nota. Quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is sought. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1262" xml:space="preserve"> +datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is given. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1263" xml:space="preserve"> +Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is given. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1264" xml:space="preserve"> +Pro exemplo ad resolutionem. +<lb/>[<emph style="it">tr: +According to this example for the solution. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1265" xml:space="preserve"> +In numeris. Sit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. 8. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. 27 +<lb/>[<emph style="it">tr: +In numbers, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mn>8</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>2</mn><mn>7</mn></mstyle></math>. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1266" xml:space="preserve"> +Hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>1</mn><mn>7</mn><mo>,</mo><mn>9</mn><mn>4</mn><mn>4</mn><mo>=</mo><mn>2</mn><mn>7</mn><mo>,</mo><mn>7</mn><mn>5</mn><mn>5</mn><mo>,</mo><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +That is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>1</mn><mn>7</mn><mn>9</mn><mn>4</mn><mn>4</mn><mo>=</mo><mn>2</mn><mn>7</mn><mn>7</mn><mn>5</mn><mn>5</mn><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1267" xml:space="preserve"> +Limites radicum. +<lb/>[<emph style="it">tr: +Limits of the roots. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f406v" o="406v" n="813"/> +<pb file="add_6782_f407" o="407" n="814"/> +<div xml:id="echoid-div255" type="page_commentary" level="2" n="255"> +<p> +<s xml:id="echoid-s1268" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1268" xml:space="preserve"> +On this page Harriot begins a general treatment of equations of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>d</mi><mi>a</mi><mi>a</mi></mstyle></math>, +with no cube or square term. +In order to preserve dimensions, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math> is used as a placeholder for a general 4-dimensional quantity; +similarly <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi><mi>d</mi></mstyle></math> is a placeholder for a 3-dimensional quantity, not necessarily a cube. +All coefficients are assumed to be positive. +Equations of this kind have two positive roots or none at all, depending on the size of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. <lb/> +For Harriot's derivation of the canonical form for unequal roots, see Add MS 6783, f. 174. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head166" xml:space="preserve" xml:lang="lat"> +c.11.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1270" xml:space="preserve"> +prob. 19. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex. +<lb/>[<emph style="it">tr: +Problem 19. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1271" xml:space="preserve"> +Species canonica <lb/> +ad radices inæquales. +<lb/>[<emph style="it">tr: +Canonical form for unequal roots. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1272" xml:space="preserve"> +nam: Si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>. erit: +<lb/>[<emph style="it">tr: +for if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>+</mo><mi>b</mi></mstyle></math> then: +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1273" xml:space="preserve"> +et ita est: +<lb/>[<emph style="it">tr: +and so it is. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1274" xml:space="preserve"> +Si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math>. erit: +<lb/>[<emph style="it">tr: +If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math> then: +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1275" xml:space="preserve"> +est enim. +<lb/>[<emph style="it">tr: +Indeed it is so. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1276" xml:space="preserve"> +est igitur +<lb/>[<emph style="it">tr: +Therefore it is so +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1277" xml:space="preserve"> +Species ad radices <lb/> +æquales. +<lb/>[<emph style="it">tr: +The case of equal roots. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1278" xml:space="preserve"> +Sunt continue proportionalia. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>. <lb/> +et: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mi>c</mi><mo>,</mo><mi>b</mi><mi>b</mi><mi>c</mi><mi>c</mi><mo>,</mo><mi>b</mi><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +There are continued proportionals <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1279" xml:space="preserve"> +Sit, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> minor radix. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, maior. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> the smaller root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> the larger. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1280" xml:space="preserve"> +Dico etiam quod: +<lb/>[<emph style="it">tr: +I also say that: +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1281" xml:space="preserve"> +Est enim. Est igitur. +<lb/>[<emph style="it">tr: +Indeed it is. Therefore is is so. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1282" xml:space="preserve"> +Ergo, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mo maxsize="1">[</mo></msqrt><mn>3</mn><mo maxsize="1">]</mo><mrow><mi>d</mi><mi>d</mi><mi>d</mi></mrow><mo>></mo><mi>a</mi></mstyle></math>. Hoc est qualibet radice. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mo maxsize="1">[</mo></msqrt><mn>3</mn><mo maxsize="1">]</mo><mrow><mi>d</mi><mi>d</mi><mi>d</mi></mrow><mo>></mo><mi>a</mi></mstyle></math>; this is so whatever the root. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f407v" o="407v" n="815"/> +<pb file="add_6782_f408" o="408" n="816"/> +<div xml:id="echoid-div256" type="page_commentary" level="2" n="256"> +<p> +<s xml:id="echoid-s1283" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1283" xml:space="preserve"> +Following from the general treatment in Add MS 6782, f. 410, f. 109, of avulsed cubics with no linear term, +Harriot here solves the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>4</mn><mn>3</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>5</mn><mn>7</mn><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> for both roots (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn><mn>0</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>4</mn><mn>5</mn></mstyle></math>). +He also shows how either root may be obtained from the other. <lb/> +The equation is taken from Problem 18 of Viète, +<emph style="it">De numerosa potestatum ad exegesin resolutione</emph>. +Viète gave rules for the relationship between the two roots but did not explain how he had arrived at them. <lb/> +This page demonstrates clearly two kinds of canonical forms used by Harriot in this treatise. +The first, the 'canonical form for unequal roots' is the general form of an avulsed cubic without a linear term; +in this case <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> are the two positive roots of the equation. +The second, the 'canonical form for the solution' is the form arrived at by assuming the root <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> +takes the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>0</mn><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>, that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is the first integer in the solution, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> the second. +Thus the meanings of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> in the two forms are thus quite different. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head167" xml:space="preserve" xml:lang="lat"> +c.10.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1285" xml:space="preserve"> +prob. 18. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex. +<lb/>[<emph style="it">tr: +Problem 18. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1286" xml:space="preserve"> +[<emph style="it">Note: +Here <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> are the two positive roots of the equation. + </emph>]<lb/> +Species canonica <lb/> +ad radices inæquales. +<lb/>[<emph style="it">tr: +Canonical form for unequal roots. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1287" xml:space="preserve"> +[<emph style="it">Note: +Here <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> are the first and second integers of any solution. + </emph>]<lb/> +Species canonica <lb/> +ad resolutionem. +<lb/>[<emph style="it">tr: +Canonical form for the solution. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1288" xml:space="preserve"> +Resolutio: +<lb/>[<emph style="it">tr: +Solution: +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1289" xml:space="preserve"> +Eductio lateris <lb/> +Minoris. +<lb/>[<emph style="it">tr: +Extraction of the smaller root. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1290" xml:space="preserve"> +Radix igitur minor, est 30. +<lb/>[<emph style="it">tr: +Therefore the smaller root is 30. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1291" xml:space="preserve"> +Quæratur iam maior. +<lb/>[<emph style="it">tr: +The larger root is now sought. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1292" xml:space="preserve"> +Sit minor <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. maior <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Let the smaller root be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, the larger <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1293" xml:space="preserve"> +Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. 45. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>4</mn><mn>5</mn></mstyle></math> is given. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1294" xml:space="preserve"> +Si quæratur minor radix. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +If the smaller root <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is sought, +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1295" xml:space="preserve"> +Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. 30. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mn>3</mn><mn>0</mn></mstyle></math> is given. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1296" xml:space="preserve"> +Eductio lateris <lb/> +Maioris. +<lb/>[<emph style="it">tr: +Solution and extraction of the larger root. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1297" xml:space="preserve"> +Radix igitur maior 45. +<lb/>[<emph style="it">tr: +Therefore the larger root is 45. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f408v" o="408v" n="817"/> +<pb file="add_6782_f409" o="409" n="818"/> +<div xml:id="echoid-div257" type="page_commentary" level="2" n="257"> +<p> +<s xml:id="echoid-s1298" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1298" xml:space="preserve"> +On this page Harriot compares his own method with that of Viète, in Problem 18 of +<emph style="it">De numerosa potestatum ad exegesin resolutione</emph>, +showing that the two methods are essentially the same. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head168" xml:space="preserve" xml:lang="lat"> +c.9.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1300" xml:space="preserve"> +prob. 18. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex. +<lb/>[<emph style="it">tr: +Problem 18. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1301" xml:space="preserve"> +Species canonica <lb/> +ad radices inæquales. +<lb/>[<emph style="it">tr: +Canonical form for unequal roots. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1302" xml:space="preserve"> +Aliter quam supra. +<lb/>[<emph style="it">tr: +Another way from that above. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1303" xml:space="preserve"> +si una radix sit nota, <lb/> +altera erit cognita. +<lb/>[<emph style="it">tr: +If one root is known, the other will be known. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1304" xml:space="preserve"> +tres continue proportionales. <lb/> +prima et secunda <lb/> +secunda et tertia. <lb/> +tertia. <lb/> +prima. +<lb/>[<emph style="it">tr: +Three continued proportionals. <lb/> +first and second <lb/> +second and third <lb/> +third <lb/> +first +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1305" xml:space="preserve"> +sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> nota. Quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is sought. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1306" xml:space="preserve"> +sint continue proportionales. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>e</mi><mi>e</mi></mrow><mrow><mi>f</mi></mrow></mfrac></mstyle></math> +<lb/>[<emph style="it">tr: +Let there be continued proportionals <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>e</mi><mi>e</mi></mrow><mrow><mi>f</mi></mrow></mfrac></mstyle></math>.] +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1307" xml:space="preserve"> +Datur igitur. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> is given. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1308" xml:space="preserve"> +Datur igitur. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is given. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1309" xml:space="preserve"> +Sed in Vieta <lb/> +iisdem præmissis. +<lb/>[<emph style="it">tr: +But in Viète from the same premises. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1310" xml:space="preserve"> +[<emph style="it">Note: +The other sheet referred to here is Add MS 6782, f. 410. + </emph>]<lb/> +eadem quæ altera charta. +<lb/>[<emph style="it">tr: +The same as in the other sheet. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1311" xml:space="preserve"> +sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> nota. Quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is sought. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1312" xml:space="preserve"> +Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> is given. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1313" xml:space="preserve"> +datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is given. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1314" xml:space="preserve"> +sed in Vieta +<lb/>[<emph style="it">tr: +But in Viete +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1315" xml:space="preserve"> +Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi><mo>=</mo><mo>-</mo><mi>g</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>b</mi></mstyle></math>. <lb/> +eadem quæ nostra in altera charta. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi><mo>=</mo><mo>-</mo><mi>g</mi><mi>b</mi><mo>=</mo><mi>b</mi><mi>b</mi></mstyle></math>, the same as mine in the other sheet. +</emph>]<lb/> +[<emph style="it">Note: +The other sheet referred to here is Add MS 6782, f. 410. + </emph>]<lb/> +</s> +</p> +<pb file="add_6782_f409v" o="409v" n="819"/> +<pb file="add_6782_f410" o="410" n="820"/> +<div xml:id="echoid-div258" type="page_commentary" level="2" n="258"> +<p> +<s xml:id="echoid-s1316" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1316" xml:space="preserve"> +On this page, Harriot continues his general treatment of equations of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, +begun on the previous page. <lb/> +The numerical example at the end, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>4</mn><mn>3</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>5</mn><mn>7</mn><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, is from Problem 18 of Viète, +<emph style="it">De numerosa potestatum ad exegesin resolutione</emph>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head169" xml:space="preserve" xml:lang="lat"> +c.8.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1318" xml:space="preserve"> +prob. 18. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex. +<lb/>[<emph style="it">tr: +Problem 18. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1319" xml:space="preserve"> +Species canonica <lb/> +ad radices inæquales. +<lb/>[<emph style="it">tr: +Canonical form for unequal roots. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1320" xml:space="preserve"> +Si una radix sit nota, <lb/> +altera erit cognita. +<lb/>[<emph style="it">tr: +If one root is known, the other will be known. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1321" xml:space="preserve"> +</s> +<lb/> +<s xml:id="echoid-s1322" xml:space="preserve"> +Dabitur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> will be given. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1323" xml:space="preserve"> +Datur igitur, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is given. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1324" xml:space="preserve"> +sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> nota. Quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is sought. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1325" xml:space="preserve"> +Dabitur ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> will be given. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1326" xml:space="preserve"> +Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is given. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1327" xml:space="preserve"> +pro exemplo ad resolutionem. +<lb/>[<emph style="it">tr: +According to this example for the solution. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1328" xml:space="preserve"> +In numeris sit. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. 30. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. 45. +<lb/>[<emph style="it">tr: +In numbers, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mn>3</mn><mn>0</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>4</mn><mn>5</mn></mstyle></math>. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1329" xml:space="preserve"> +Hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>4</mn><mn>3</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>5</mn><mn>7</mn><mo>,</mo><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +That is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>4</mn><mn>3</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>5</mn><mn>7</mn><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p> +<s xml:id="echoid-s1330" xml:space="preserve"> +Limites radicum. +<lb/>[<emph style="it">tr: +Limits of the roots. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f410v" o="410v" n="821"/> +<pb file="add_6782_f411" o="411" n="822"/> +<div xml:id="echoid-div259" type="page_commentary" level="2" n="259"> +<p> +<s xml:id="echoid-s1331" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1331" xml:space="preserve"> +On this page Harriot begins a general treatment of equations of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, +with no linear term. All coefficients are assumed to be positive. +Equations of this kind have two positive roots or none at all, depending on the size of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. <lb/> +For Harriot's derivation of the canonical form for unequal roots, see Add MS 6783, f. 181. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head170" xml:space="preserve" xml:lang="lat"> +c.7.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1333" xml:space="preserve"> +prob. 18. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex. +<lb/>[<emph style="it">tr: +Problem 18. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1334" xml:space="preserve"> +Species canonica <lb/> +ad radices inæquales. +<lb/>[<emph style="it">tr: +Canonical form for unequal roots. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1335" xml:space="preserve"> +Nam: Si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>. <lb/> +erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>b</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mi>b</mi><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>. Et ita est: +<lb/>[<emph style="it">tr: +For if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>b</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mi>b</mi><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>; and so it is. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1336" xml:space="preserve"> +Si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math>. <lb/> +erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mi>c</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>c</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>c</mi><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>. est enim. +<lb/>[<emph style="it">tr: +If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mi>c</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>c</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>c</mi><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>; indeed it is. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1337" xml:space="preserve"> +est igitur +<lb/>[<emph style="it">tr: +Therefore it is so. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1338" xml:space="preserve"> +Species ad radices <lb/> +æquales. +<lb/>[<emph style="it">tr: +The case of equal roots. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1339" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math> sunt continue proportionalia. +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math> are in continued proportion. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1340" xml:space="preserve"> +Dico etiam quod: +<lb/>[<emph style="it">tr: +I say also that: +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1341" xml:space="preserve"> +ponatur: est igitur: +<lb/>[<emph style="it">tr: +This supposed, then +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1342" xml:space="preserve"> +est enim, est igitur. +<lb/>[<emph style="it">tr: +Indded it is; therefore it is so. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1343" xml:space="preserve"> +est igitur: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>></mo><mi>a</mi></mstyle></math>. hoc est qualibet <lb/> +radice. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>></mo><mi>a</mi></mstyle></math>; this is so whatever the root. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f411v" o="411v" n="823"/> +<pb file="add_6782_f412" o="412" n="824"/> +<div xml:id="echoid-div260" type="page_commentary" level="2" n="260"> +<p> +<s xml:id="echoid-s1344" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1344" xml:space="preserve"> +Following on from Add MS 6782, f. 415, f. 414, f. 413, Harriot here solves the equation +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>5</mn><mn>5</mn><mn>5</mn><mn>2</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>3</mn><mn>1</mn><mn>0</mn><mn>4</mn><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> for the larger root (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>1</mn><mn>0</mn><mn>8</mn></mstyle></math>). +He then shows how the smaller roots (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>1</mn><mn>2</mn></mstyle></math>) may be derived from the larger one. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head171" xml:space="preserve" xml:lang="lat"> +c.6.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1346" xml:space="preserve"> +prob. 17. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex. +<lb/>[<emph style="it">tr: +Problem 17. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1347" xml:space="preserve"> +Species canonica <lb/> +ad resolutionem. +<lb/>[<emph style="it">tr: +Canonical form for the solution. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1348" xml:space="preserve"> +Resolutio <lb/> +et eductio lateris <lb/> +Maioris. +<lb/>[<emph style="it">tr: +Solution and extraction of the larger root. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1349" xml:space="preserve"> +Radix <emph style="super">igitur</emph> maior est 108. +<lb/>[<emph style="it">tr: +Therefore the larger root is 108. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1350" xml:space="preserve"> +Quæratur iam minor. +<lb/>[<emph style="it">tr: +Now there is sought the smaller root. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1351" xml:space="preserve"> +Sit maior <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. minor <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Let the larger root be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, the smaller <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1352" xml:space="preserve"> +Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. 12. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mn>1</mn><mn>2</mn></mstyle></math> is given. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f412v" o="412v" n="825"/> +<pb file="add_6782_f413" o="413" n="826"/> +<div xml:id="echoid-div261" type="page_commentary" level="2" n="261"> +<p> +<s xml:id="echoid-s1353" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1353" xml:space="preserve"> +Following from the general treatment in Add MS 6782, f. 415, f. 414, of avulsed cubics with no square term, +Harriot here solves the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>5</mn><mn>5</mn><mn>5</mn><mn>2</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>3</mn><mn>1</mn><mn>0</mn><mn>4</mn><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math> for the smaller root (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>1</mn><mn>2</mn></mstyle></math>). +He then shows how the larger roots (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>1</mn><mn>0</mn><mn>8</mn></mstyle></math>) may be derived from the smaller one. <lb/> +The equation is taken from Problem 17 of Viète, +<emph style="it">De numerosa potestatum ad exegesin resolutione</emph>. +Viète gave rules for the relationship between the two roots but did not explain how he had arrived at them. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head172" xml:space="preserve" xml:lang="lat"> +c.5.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1355" xml:space="preserve"> +prob. 17. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex. +<lb/>[<emph style="it">tr: +Problem 17. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1356" xml:space="preserve"> +Species canonica <lb/> +ad resolutionem. +<lb/>[<emph style="it">tr: +Canonical form for the solution. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1357" xml:space="preserve"> +Resolutio <lb/> +et eductio lateris <lb/> +minoris. +<lb/>[<emph style="it">tr: +Solution and extraction of the smaller root. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1358" xml:space="preserve"> +Radix igitur minor est, 12. +<lb/>[<emph style="it">tr: +Therefore the smaller root is 12. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1359" xml:space="preserve"> +Radix <emph style="super">minor</emph> sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. maior <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. <lb/> +Et quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +The smaller root is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, the larger <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>; and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is sought. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1360" xml:space="preserve"> +Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. 108. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>1</mn><mn>0</mn><mn>8</mn></mstyle></math> is given. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1361" xml:space="preserve"> +A. Poristicum +<lb/>[<emph style="it">tr: +Proof +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1362" xml:space="preserve"> +Quod: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi><mo>-</mo><mi>b</mi><mi>b</mi><mo>=</mo><mfrac><mrow><mi>x</mi><mi>x</mi><mi>z</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> +<lb/>[<emph style="it">tr: +Because <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi><mo>-</mo><mi>b</mi><mi>b</mi><mo>=</mo><mfrac><mrow><mi>x</mi><mi>x</mi><mi>z</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1363" xml:space="preserve"> +Hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +That is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi></mstyle></math>. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1364" xml:space="preserve"> +Est enim. est igitur. +<lb/>[<emph style="it">tr: +Indeed it is; therefore it is so. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1365" xml:space="preserve"> +Etiam. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi><mo>-</mo><mi>c</mi><mi>c</mi><mo>=</mo><mfrac><mrow><mi>x</mi><mi>x</mi><mi>z</mi></mrow><mrow><mi>c</mi></mrow></mfrac></mstyle></math>. +<lb/>[<emph style="it">tr: +Also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi><mo>-</mo><mi>c</mi><mi>c</mi><mo>=</mo><mfrac><mrow><mi>x</mi><mi>x</mi><mi>z</mi></mrow><mrow><mi>c</mi></mrow></mfrac></mstyle></math>. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1366" xml:space="preserve"> +hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +That is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi></mstyle></math>. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1367" xml:space="preserve"> +Est enim. est igitur. +<lb/>[<emph style="it">tr: +Indeed it is; therefore it is so. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f413v" o="413v" n="827"/> +<pb file="add_6782_f414" o="414" n="828"/> +<div xml:id="echoid-div262" type="page_commentary" level="2" n="262"> +<p> +<s xml:id="echoid-s1368" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1368" xml:space="preserve"> +On this page, Harriot continues his general treatment of equations of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, +begun on the previous page. <lb/> +The numerical example at the end, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>5</mn><mn>5</mn><mn>5</mn><mn>2</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>3</mn><mn>1</mn><mn>0</mn><mn>4</mn><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, is from Problem 17 of Viète, +<emph style="it">De numerosa potestatum ad exegesin resolutione</emph>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head173" xml:space="preserve" xml:lang="lat"> +c.4.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1370" xml:space="preserve"> +prob. 17. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex. +<lb/>[<emph style="it">tr: +Problem 17. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1371" xml:space="preserve"> +Species canonica <lb/> +ad radices inæquales. +<lb/>[<emph style="it">tr: +Canonical form for unequal roots. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1372" xml:space="preserve"> +Si una radix sit nota, <lb/> +altera erit cognita. +<lb/>[<emph style="it">tr: +If one root is known, the other will be known. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1373" xml:space="preserve"> +Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> nota. Quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is sought. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1374" xml:space="preserve"> +Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is given. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1375" xml:space="preserve"> +vel: +<lb/>[<emph style="it">tr: +or: +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1376" xml:space="preserve"> +Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is given. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1377" xml:space="preserve"> +sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> nota. Quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is sought. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1378" xml:space="preserve"> +Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is given. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1379" xml:space="preserve"> +vel: +<lb/>[<emph style="it">tr: +or: +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1380" xml:space="preserve"> +Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is given. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1381" xml:space="preserve"> +Pro exemplo ad resolutionem. +<lb/>[<emph style="it">tr: +According to this example for the solution. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1382" xml:space="preserve"> +In numeris. Sit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. 12. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. 108. +<lb/>[<emph style="it">tr: +In numbers, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mn>1</mn><mn>2</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>1</mn><mn>0</mn><mn>8</mn></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1383" xml:space="preserve"> +Limites radicum. +<lb/>[<emph style="it">tr: +Limits of the roots. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f414v" o="414v" n="829"/> +<pb file="add_6782_f415" o="415" n="830"/> +<div xml:id="echoid-div263" type="page_commentary" level="2" n="263"> +<p> +<s xml:id="echoid-s1384" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1384" xml:space="preserve"> +On this page Harriot begins a general treatment of equations of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>, +with no square term. To preserve dimensions, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math> is used as a placeholder for a general 3-dimensional quantity; +similarly <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>d</mi></mstyle></math> is a placeholder for a 2-dimensional quantity, not necessarily a square. +All coefficients are assumed to be positive. +Equations of this kind have two positive roots or none at all, depending on the size of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. <lb/> +For Harriot's derivation of the canonical form for unequal roots, see Add MS 6783, f. 181. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head174" xml:space="preserve" xml:lang="lat"> +c.3.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1386" xml:space="preserve"> +prob. 17. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex. +<lb/>[<emph style="it">tr: +Problem 17. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1387" xml:space="preserve"> +Species canonica <lb/> +ad radices inæquales. +<lb/>[<emph style="it">tr: +Canonical form for unequal roots. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1388" xml:space="preserve"> +Nam: Si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>. erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>b</mi><mi>b</mi></mstyle></math>. Et ita est: +<lb/>[<emph style="it">tr: +For if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>b</mi><mi>b</mi></mstyle></math>; and so it is. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1389" xml:space="preserve"> +Si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math>. erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>. est enim. +<lb/>[<emph style="it">tr: +If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>c</mi><mi>c</mi><mi>c</mi></mstyle></math>; and indeed it is so. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1390" xml:space="preserve"> +est igitur <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore it is so, that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1391" xml:space="preserve"> +Species ad radices <lb/> +æquales. +<lb/>[<emph style="it">tr: +The case of equal roots. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1392" xml:space="preserve"> +si: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math> <lb/> +erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>=</mo><mi>x</mi><mi>x</mi><mi>z</mi></mstyle></math>. +</emph>]<lb/> +[<emph style="it">Note: +If the cubic has two positive roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, then the third root must be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mi>b</mi><mo>-</mo><mi>c</mi></mstyle></math>. +Hence the product of the roots is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mo maxsize="1">(</mo><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo maxsize="1">)</mo></mstyle></math>. + </emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1393" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math>. sunt continue <lb/> +proportionalia. +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math>, are in continued proportion. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1394" xml:space="preserve"> +sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, minor radix. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, maior. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the smaller root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> the larger. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1395" xml:space="preserve"> +Ergo qualibet radix non <lb/> +habet plures figuras <lb/> +quam sunt in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>d</mi><mi>d</mi></mrow></msqrt></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore whatever the root, it has no more figures than are in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>d</mi><mi>d</mi></mrow></msqrt></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f415v" o="415v" n="831"/> +<pb file="add_6782_f416" o="416" n="832"/> +<div xml:id="echoid-div264" type="page_commentary" level="2" n="264"> +<p> +<s xml:id="echoid-s1396" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1396" xml:space="preserve"> +On the previous page, Add MS 6782, f. 417, Harriot discussed equations of the general form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>. +On that page, as an example, he calculated limits for the two positive roots of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>9</mn><mn>2</mn><mn>6</mn><mn>1</mn><mo>=</mo><mn>3</mn><mn>7</mn><mn>0</mn><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>. +On this page he solves the same equation fully for both roots. <lb/> +The equation is taken from Problem 16 of Viète, +<emph style="it">De numerosa potestatum ad exegesin resolutione</emph>. +Viète gave rules for the relationship between the two roots but did not explain how he had arrived at them. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head175" xml:space="preserve" xml:lang="lat"> +c.2.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1398" xml:space="preserve"> +prob. 16. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex. +<lb/>[<emph style="it">tr: +Problem 16. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1399" xml:space="preserve"> +Species canonica <lb/> +ad resolutione. +<lb/>[<emph style="it">tr: +Canonical form for the solution. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1400" xml:space="preserve"> +Resolutio, <lb/> +et eductio <lb/> +lateris minoris +<lb/>[<emph style="it">tr: +Solution and extraction of the smaller root. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1401" xml:space="preserve"> +Divisor +<lb/>[<emph style="it">tr: +Divisor +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1402" xml:space="preserve"> +Ergo. 27. latus minus. <lb/> +ergo: latus maius. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mn>7</mn><mn>0</mn><mo>-</mo><mn>2</mn><mn>7</mn><mo>=</mo><mn>3</mn><mn>4</mn><mn>3</mn></mstyle></math>. <lb/> +vel: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>9</mn><mn>2</mn><mn>6</mn><mn>1</mn></mrow><mrow><mn>2</mn><mn>7</mn></mrow></mfrac><mo>=</mo><mn>3</mn><mn>4</mn><mn>3</mn></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore 27 is the smaller root. <lb/> +Therefore the larger root is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mn>7</mn><mn>0</mn><mo>-</mo><mn>2</mn><mn>7</mn><mo>=</mo><mn>3</mn><mn>4</mn><mn>3</mn></mstyle></math>. <lb/> +or: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>9</mn><mn>2</mn><mn>6</mn><mn>1</mn></mrow><mrow><mn>2</mn><mn>7</mn></mrow></mfrac><mo>=</mo><mn>3</mn><mn>4</mn><mn>3</mn></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1403" xml:space="preserve"> +Eductio <lb/> +lateris maioris. +<lb/>[<emph style="it">tr: +Extraction of the larger root. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1404" xml:space="preserve"> +Divisor. +<lb/>[<emph style="it">tr: +Divisor. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1405" xml:space="preserve"> +Divisor. +<lb/>[<emph style="it">tr: +Divisor. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1406" xml:space="preserve"> +Ergo. latus maius. 343 <lb/> +ergo laus minor. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mn>7</mn><mn>0</mn><mo>-</mo><mn>3</mn><mn>4</mn><mn>3</mn><mo>=</mo><mn>2</mn><mn>7</mn></mstyle></math>. <lb/> +vel. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>9</mn><mn>2</mn><mn>6</mn><mn>1</mn></mrow><mrow><mn>3</mn><mn>4</mn><mn>3</mn></mrow></mfrac><mo>=</mo><mn>2</mn><mn>7</mn></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore the greater root is 343. <lb/> +therefore the smaller root is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mn>7</mn><mn>0</mn><mo>-</mo><mn>3</mn><mn>4</mn><mn>3</mn><mo>=</mo><mn>2</mn><mn>7</mn></mstyle></math>. <lb/> +or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>9</mn><mn>2</mn><mn>6</mn><mn>1</mn></mrow><mrow><mn>3</mn><mn>4</mn><mn>3</mn></mrow></mfrac><mo>=</mo><mn>2</mn><mn>7</mn></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f416v" o="416v" n="833"/> +<pb file="add_6782_f417" o="417" n="834"/> +<div xml:id="echoid-div265" type="page_commentary" level="2" n="265"> +<p> +<s xml:id="echoid-s1407" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1407" xml:space="preserve"> +This is the first of a set of 18 pages on extracting roots of avulsed equations, +that is, equations containing three terms, in which the term of highest degree is subtracted +(torn away, or 'avulsed') from the term of next higest degree. +Such equations have two positive roots. It is therefore important to know the relative sizes of the roots +before beginning extraction by numerical methods. This is the problem Harriot investigates in this section. <lb/> +The work is closely based on Problems 16 to 20 in +Viète, <emph style="it">De numerosa potestatum ad exegesin resolutione</emph>, 1600. +Viète gave rules for finding the second positive root once the first is known, but without explanation. +In this section, Harriot fills in the missing details, showing how the two positive roots +are related to the coefficients of the original equation and to each other. <lb/> +For a general explanation of the method of extraction see Add MS 6782, f. 399. +For further discussion see See Stedall 2003, 87–123 and 294, and Stedall 2011, 29–33. <lb/> +This first page of Section c is transcribed in full, but for subsequent pages, +only phrases and sentences are transcribed, not the calculations or the single words used in them. <lb/> +For another version of the first page see Add MS 6783, f. 62v, f. 62. <lb/> +For Harriot's derivation of the canonical form for unequal roots, see Add MS 6783, f. 183. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head176" xml:space="preserve" xml:lang="lat"> +c.1.) De numerosa potestatum resolutione. +<lb/>[<emph style="it">tr: +On the numerical resolution of powers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1409" xml:space="preserve"> +prob. 16. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, duplex. +<lb/>[<emph style="it">tr: +Problem 16. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> twofold. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1410" xml:space="preserve"> +Species canonica <lb/> +ad radices inæquales. +<lb/>[<emph style="it">tr: +Canonical form for unequal roots. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1411" xml:space="preserve"> +Nam: Si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>. erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>b</mi></mstyle></math>. et ita est. +<lb/>[<emph style="it">tr: +For if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>b</mi></mstyle></math>; and so it is. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1412" xml:space="preserve"> +Si, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math>. erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mo>-</mo><mi>c</mi><mi>c</mi></mstyle></math>. est enim. +<lb/>[<emph style="it">tr: +If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>=</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mo>-</mo><mi>c</mi><mi>c</mi></mstyle></math>; indeed it is so. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1413" xml:space="preserve"> +est igitur <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +therefore it is so, that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>c</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1414" xml:space="preserve"> +Species ad radices <lb/> +æquales. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>=</mo><mi>b</mi><mi>a</mi><mo>+</mo><mi>b</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math> +<lb/>[<emph style="it">tr: +Canonical form for equal roots. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1415" xml:space="preserve"> +Vel: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>=</mo><mn>2</mn><mi>b</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>=</mo><mn>2</mn><mi>b</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1416" xml:space="preserve"> +si: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi><mo>=</mo><mi>d</mi></mstyle></math> <lb/> +erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>=</mo><mi>x</mi><mi>z</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +if: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi><mo>=</mo><mi>d</mi></mstyle></math> <lb/> +then: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>=</mo><mi>x</mi><mi>z</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1417" xml:space="preserve"> +Sunt in ratione inæqualitatis: <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, be in unequal ratio. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1418" xml:space="preserve"> +sit, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> radix minor. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, maior. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>b</mi><mo><</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo><</mo><mn>2</mn><mi>c</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo><</mo><mfrac><mrow><mi>b</mi><mo>+</mo><mi>c</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo><</mo><mi>c</mi></mstyle></math>. <lb/> +Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo><</mo><mfrac><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo><</mo><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the smaller root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> the larger. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>b</mi><mo><</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo><</mo><mn>2</mn><mi>c</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo><</mo><mfrac><mrow><mi>b</mi><mo>+</mo><mi>c</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo><</mo><mi>c</mi></mstyle></math>. <lb/> +Therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo><</mo><mfrac><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo><</mo><mi>c</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1419" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo><</mo><mi>b</mi><mi>c</mi><mo><</mo><mi>c</mi><mi>c</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo><</mo><msqrt><mrow><mi>b</mi><mi>c</mi></mrow></msqrt><mo><</mo><mi>c</mi></mstyle></math>. <lb/> +Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo><</mo><msqrt><mrow><mi>x</mi><mi>z</mi></mrow></msqrt><mo><</mo><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo><</mo><mi>b</mi><mi>c</mi><mo><</mo><mi>c</mi><mi>c</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo><</mo><msqrt><mrow><mi>b</mi><mi>c</mi></mrow></msqrt><mo><</mo><mi>c</mi></mstyle></math>. <lb/> +Therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo><</mo><msqrt><mrow><mi>x</mi><mi>z</mi></mrow></msqrt><mo><</mo><mi>c</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1420" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>,</mo><mn>2</mn><mo>,</mo><mi>x</mi><mi>z</mi><mo>:</mo><mn>1</mn><mo>,</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mi>z</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math> <lb/> +Dico quod: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo><</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mi>z</mi></mrow><mrow><mi>d</mi></mrow></mfrac><mo><</mo><mi>c</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi><mo><</mo><mn>2</mn><mo>,</mo><mi>x</mi><mi>z</mi><mo><</mo><mi>c</mi><mi>d</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo><</mo><mn>2</mn><mi>b</mi><mi>c</mi><mo><</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi></mstyle></math> <lb/> +est enim. est igitur +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>:</mo><mn>2</mn><mi>x</mi><mi>z</mi><mo>=</mo><mn>1</mn><mo>:</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mi>z</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math> <lb/> +I say that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo><</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mi>z</mi></mrow><mrow><mi>d</mi></mrow></mfrac><mo><</mo><mi>c</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi><mo><</mo><mn>2</mn><mi>x</mi><mi>z</mi><mo><</mo><mi>c</mi><mi>d</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo><</mo><mn>2</mn><mi>b</mi><mi>c</mi><mo><</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi></mstyle></math> <lb/> +indeed it is; therefore it is so. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1421" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math> <lb/> +Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>a</mi><mo>></mo><mi>a</mi><mi>a</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>></mo><mi>a</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math> <lb/> +Therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>a</mi><mo>></mo><mi>a</mi><mi>a</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>></mo><mi>a</mi></mstyle></math>. +</emph>]<lb/> +</s> +<s xml:id="echoid-s1422" xml:space="preserve"> +Hoc est qualibet <lb/> +radice. +<lb/>[<emph style="it">tr: +This is so whatever the root. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1423" xml:space="preserve"> +Ergo [???] radix habet plures <lb/> +figuras quam sunt in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore the [???] root has more figures than are in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1424" xml:space="preserve"> +Si una radix sit nota: <lb/> +altera erit cognita. +<lb/>[<emph style="it">tr: +If one root is known, the other will be known. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1425" xml:space="preserve"> +Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> nota. quæratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is sought. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1426" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>=</mo><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> <lb/> +ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>b</mi><mo>=</mo><mi>c</mi></mstyle></math>. <lb/> +vel: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi><mo>=</mo><mi>b</mi><mi>c</mi></mstyle></math> <lb/> +ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>x</mi><mi>z</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>=</mo><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> <lb/> +therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>b</mi><mo>=</mo><mi>c</mi></mstyle></math>. <lb/> +or: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi><mo>=</mo><mi>b</mi><mi>c</mi></mstyle></math> <lb/> +therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>x</mi><mi>z</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mi>c</mi></mstyle></math>. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1427" xml:space="preserve"> +Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> nota. quæratur, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>c</mi><mo>=</mo><mi>b</mi></mstyle></math> <lb/> +et: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>x</mi><mi>z</mi></mrow><mrow><mi>c</mi></mrow></mfrac><mo>=</mo><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> be known, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is sought. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>c</mi><mo>=</mo><mi>b</mi></mstyle></math> <lb/> +and: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>x</mi><mi>z</mi></mrow><mrow><mi>c</mi></mrow></mfrac><mo>=</mo><mi>b</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1428" xml:space="preserve"> +Pro exemplo ad resolutionem. +<lb/>[<emph style="it">tr: +According to this example for the solution. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1429" xml:space="preserve"> +In numeris sit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. 27. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> 343. +<lb/>[<emph style="it">tr: +In numbers, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mn>2</mn><mn>7</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>3</mn><mn>4</mn><mn>3</mn></mstyle></math>. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1430" xml:space="preserve"> +Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>7</mn><mo>×</mo><mn>3</mn><mn>4</mn><mn>3</mn><mo>=</mo><mn>2</mn><mn>7</mn><mo>,</mo><mi>a</mi><mo>+</mo><mn>3</mn><mn>4</mn><mn>3</mn><mo>,</mo><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>. <lb/> +Hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>9</mn><mn>2</mn><mn>6</mn><mn>1</mn><mo>=</mo><mn>3</mn><mn>7</mn><mn>0</mn><mo>,</mo><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>7</mn><mo>×</mo><mn>3</mn><mn>4</mn><mn>3</mn><mo>=</mo><mn>2</mn><mn>7</mn><mi>a</mi><mo>+</mo><mn>3</mn><mn>4</mn><mn>3</mn><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>. <lb/> +That is: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>9</mn><mn>2</mn><mn>6</mn><mn>1</mn><mo>=</mo><mn>3</mn><mn>7</mn><mn>0</mn><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1431" xml:space="preserve"> +Limites radicum ex præcedentibus. +<lb/>[<emph style="it">tr: +The limits of the roots from what has gone before. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f417v" o="417v" n="835"/> +<p> +<s xml:id="echoid-s1432" xml:space="preserve"> +Johan Dycker <lb/> +Johan +</s> +</p> +<pb file="add_6782_f418" o="418" n="836"/> +<pb file="add_6782_f418v" o="418v" n="837"/> +<pb file="add_6782_f419" o="419" n="838"/> +<div xml:id="echoid-div266" type="page_commentary" level="2" n="266"> +<p> +<s xml:id="echoid-s1433" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1433" xml:space="preserve">The referenceon this page is to Proposition 15 from Chapter 19 of Viète's +<emph style="it">Variorum responsorum liber VIII</emph> (1593). +</s> +<lb/> +<quote xml:lang="lat"> +XV. Datis tribus lateribus, dantur anguli. +</quote> +<lb/> +<quote> +Given three sides, the angles are given. +</quote> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head177" xml:space="preserve" xml:lang="lat"> +Lemmata quædam <lb/> +ad praxin prop. 15 <lb/> +Vieta lib. resp. 8. <lb/> +pag. 35 +<lb/>[<emph style="it">tr: +Certain lemmas for carrying out Proposition 15, Viète, Responsorum liber VIII, page 35. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1435" xml:space="preserve"> +1. Duæ peripheriæ sigillatim minores quadranti, et earum complementa: <lb/> +æqualem habent differentiam. <lb/> +sint duæ peripheriæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>, differentia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> <lb/> +complementa earum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>o</mi></mstyle></math>, differentia etiam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +1. Two arcs each less than a quadrant, and their complements, have equal differences. <lb/> +Let the two arcs be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> abd <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>, with difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. <lb/> +Their complements are <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi></mstyle></math> adn <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>o</mi></mstyle></math>, also with difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1436" xml:space="preserve"> +2. Complementa aggregati duarum peripheriarum sigillatim minorum quadranti; et <lb/> +aggregatum illarum complementorum: sunt æqualia. <lb/> +aggregatum ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math>. complementum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>d</mi></mstyle></math>. <lb/> +complementum peripheriæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi></mstyle></math>, hoc est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>o</mi></mstyle></math>. <lb/> +complementum peripheriæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>, est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>o</mi></mstyle></math>. <lb/> +Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>o</mi><mo>+</mo><mi>o</mi><mi>c</mi></mstyle></math> est aggregatum complementorum <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>i</mi></mstyle></math> sunt æquales. <lb/> +Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>o</mi><mi>d</mi></mstyle></math> est aggregatum complementorum et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>d</mi></mstyle></math> complementum aggregat. +<lb/>[<emph style="it">tr: +2. The complements of the sum of two arcs each less than a quadrant, +and the sum of those complements, are equal. <lb/> +Therefore if the sum is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math> the complement is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>d</mi></mstyle></math>. <lb/> +The complement of the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi></mstyle></math>, that is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>o</mi></mstyle></math>. <lb/> +The complement of the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>o</mi></mstyle></math>. <lb/> +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>o</mi><mo>+</mo><mi>o</mi><mi>c</mi></mstyle></math> is the sum of the complements. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>i</mi></mstyle></math> are equal. <lb/> +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>o</mi><mo>+</mo><mi>o</mi><mi>c</mi></mstyle></math> is the sum of the complements. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>i</mi></mstyle></math> are equal. <lb/> +Therfore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>o</mi><mi>d</mi></mstyle></math> is the sum of the coplements and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>d</mi></mstyle></math> is the complement of the sum. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f419v" o="419v" n="839"/> +<pb file="add_6782_f420" o="420" n="840"/> +<div xml:id="echoid-div267" type="page_commentary" level="2" n="267"> +<p> +<s xml:id="echoid-s1437" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1437" xml:space="preserve"> +This page continues Harriot's work from Add MS 6787, f. 61, and Add MS 782, f. 422. +on Viète's statement of 'Syntomon'. <lb/> +The third case is where one angle is greater than a right angle, the other less. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head178" xml:space="preserve" xml:lang="lat"> +<foreign xml:lang="gre">Syntomon</foreign> Secundo. +<lb/>[<emph style="it">tr: +Syntomon, third case. +</emph>]<lb/> +</head> +<pb file="add_6782_f420v" o="420v" n="841"/> +<pb file="add_6782_f421" o="421" n="842"/> +<div xml:id="echoid-div268" type="page_commentary" level="2" n="268"> +<p> +<s xml:id="echoid-s1439" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1439" xml:space="preserve"> +This page continues Harriot's work from Add MS 6787, f. 61, +on Viète's statement of 'Syntomon'. <lb/> +The second case is where both angles are greater than a right angle. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head179" xml:space="preserve" xml:lang="lat"> +Vieta lib. 8. resp. <lb/> +pag. 39. <lb/> +<foreign xml:lang="gre">Syntomon</foreign> Secundo. +<lb/>[<emph style="it">tr: +Viète, Responsorum liber VIII, page 39, Syntomon, second case. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1441" xml:space="preserve"> +Interpretatio. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> una peripheria <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> altera <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi><mo>=</mo><mi>a</mi><mi>d</mi></mstyle></math> <lb/> +Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math> differentia. +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1442" xml:space="preserve"> +1. Si duo rectangula fuerint sigillatim applicata ad <lb/> +2. sinum totum; unum, <emph style="super">*</emph> <emph style="ul">duorum sinum</emph>, +quorum utraque peripheriæ <lb/> +sunt quadranti minores; alterum <emph style="super">+</emph> <emph style="ul">Maiorem periphe-</emph> <lb/> +<emph style="ul">riæ sinum</emph> complementarum: Duæ latitudines <lb/> +oriundæ component sinum complementi differentiæ <lb/> +peripheriarum. +<lb/>[<emph style="it">tr: +1, 2. If two rectangles are each applied to the whole sine, +one <emph style="super">*</emph>of two sines, of which either arc is less than the quadrant, +the other <emph style="super">+</emph> greater than the sine of the complement of the arc, +then the two latitudes arising are composed of the sine of the complement of the differences of the arcs. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1443" xml:space="preserve"> +* sub duobus sinibus +<lb/>[<emph style="it">tr: +* under two sines +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1444" xml:space="preserve"> ++ sub illarum <lb/> +sinibus +<lb/>[<emph style="it">tr: +under the sines of them +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1445" xml:space="preserve"> +3. Si duo rectangula fuerint sigillatimm applicata ad <lb/> +4. sinum totum; unum, <emph style="super">*</emph> <emph style="ul">duorum sinum</emph>, +quorum peripheriæ sunt <lb/> +affectionis inter se diversæ; alterum, <emph style="ul">illarum peripheriæ</emph> <lb/> +<emph style="ul">sinum</emph> complementarum: Duæ latitudines oriundæ <lb/> +component sinum complementi aggregati peripheriæ. +<lb/>[<emph style="it">tr: +3, 4. If two rectangles are each applied to the whole sine, +one <emph style="super">*</emph>of two sines, of which the relationship to the arc is different, +the other the complements of the sines of those arcs, +then the two latitudes arising are composed of the sine of the complement of the sum of the arcs. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f421v" o="421v" n="843"/> +<pb file="add_6782_f422" o="422" n="844"/> +<div xml:id="echoid-div269" type="page_commentary" level="2" n="269"> +<p> +<s xml:id="echoid-s1446" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1446" xml:space="preserve">The reference on this page is to Proposition 20 from Chapter 19 of Viète's +<emph style="it">Variorum responsorum liber VIII</emph> (1593). +</s> +<lb/> +<quote xml:lang="lat"> +XX. <lb/> +Trianguli cujuslibet sphærici. <lb/> +Datis angulis duobus, & latere quod iis adjacent, datur angulus reliquus. +</quote> +<lb/> +<quote> +Given two angles and the side adjacent to them, the other angle is given. +</quote> +<lb/> +<s xml:id="echoid-s1447" xml:space="preserve"> +Viète described four possible cases for this proposition; Harriot claims that he has missed some. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head180" xml:space="preserve" xml:lang="lat"> +Vieta. resp. lib. 8. <lb/> +pag. 38. b. <lb/> +Triangula ambigua +<lb/>[<emph style="it">tr: +Viète, Responsorum liber VIII, page 38v. <lb/> +Ambiguous triangles. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1449" xml:space="preserve"> +Iisdem positis <lb/> +Quadrati etiam: <lb/> +duo casus omissi a Vieta +<lb/>[<emph style="it">tr: +The same things being supposed, the quadrants are also: <lb/> +Two cases missed by Viète. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1450" xml:space="preserve"> +Inde <lb/> +Habendis angulus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> <lb/> +[???] duorum reliquorum <lb/> +angulorum. +<lb/>[<emph style="it">tr: +Having the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, hence the other two angles. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1451" xml:space="preserve"> +Quadranti etiam: <lb/> +duo casus omissi a Vieta +<lb/>[<emph style="it">tr: +The quadrants are also: <lb/> +Two cases missed by Viète. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1452" xml:space="preserve"> +Nota <lb/> +Obliangulorum sphæricorum <lb/> +duodecim sunt [???] <lb/> +quarum per perficuntu per <lb/> +syntomon et [???] <emph style="st">operatione</emph> <lb/> +[etc.] +<lb/>[<emph style="it">tr: +1. Two arcs each less tham a quadrant, and their complements, have equal differences. <lb/> +Let the two arcs be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>, with difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. <lb/> +Their complements are <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>o</mi></mstyle></math>, also with difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f422v" o="422v" n="845"/> +<pb file="add_6782_f423" o="423" n="846"/> +<pb file="add_6782_f423v" o="423v" n="847"/> +<pb file="add_6782_f424" o="424" n="848"/> +<div xml:id="echoid-div270" type="page_commentary" level="2" n="270"> +<p> +<s xml:id="echoid-s1453" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1453" xml:space="preserve"> +Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of +Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). +In the 1646 edition of Viete's <emph style="it">Opera mathematica</emph> +the triangle referred to here is to be found on page 423. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head181" xml:space="preserve" xml:lang="lat"> +Vieta. resp. lib. 8. <lb/> +pag. 43. b. +<lb/>[<emph style="it">tr: +Viète, Responsorum liber VIII, page 43v. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1455" xml:space="preserve"> +Nota <lb/> +Ergo datis 2<emph style="super">bus</emph> lateribuset angulo complemento <lb/> +datur latus oppositum copendiose. +<lb/>[<emph style="it">tr: +Note <lb/> +Therefore given the two sides and the complement of the angle, the opposite side is given more briefly. +</emph>]<lb/> +</s> +</p> +<p> +<s xml:id="echoid-s1456" xml:space="preserve"> +See the papers of sines <lb/> +proportionall. +</s> +</p> +<pb file="add_6782_f424v" o="424v" n="849"/> +<pb file="add_6782_f425" o="425" n="850"/> +<pb file="add_6782_f425v" o="425v" n="851"/> +<pb file="add_6782_f426" o="426" n="852"/> +<div xml:id="echoid-div271" type="page_commentary" level="2" n="271"> +<p> +<s xml:id="echoid-s1457" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1457" xml:space="preserve"> +Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of +Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). +In the 1646 edition of Viete's <emph style="it">Opera mathematica</emph> +the triangle referred to here is to be found on page 423. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head182" xml:space="preserve" xml:lang="lat"> +Vieta. resp. lib. 8. <lb/> +pag. 43. b. <lb/> +1. Datis tribus lateribis <lb/> +quæritur angulus A. +<lb/>[<emph style="it">tr: +Viète, Responsorum liber VIII, page 43v. <lb/> +Given three sides, there is sought angle A. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1459" xml:space="preserve"> +Analogia <lb/> +Vide syntomon 2<emph style="super">o</emph>. <lb/> +Angulus quæsitis +<lb/>[<emph style="it">tr: +Ratio <lb/> +See syntomon 2. <lb/> +Angle sought. +</emph>]<lb/> +[<emph style="it">Note: +The second case of syntomon can be found on Add MS 6782, f. 421. + </emph>]<lb/> +</s> +</p> +<pb file="add_6782_f426v" o="426v" n="853"/> +<pb file="add_6782_f427" o="427" n="854"/> +<div xml:id="echoid-div272" type="page_commentary" level="2" n="272"> +<p> +<s xml:id="echoid-s1460" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1460" xml:space="preserve"> +Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of +Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). +In the 1646 edition of Viete's <emph style="it">Opera mathematica</emph> +the triangle referred to here is to be found on page 423. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head183" xml:space="preserve" xml:lang="lat"> +Vieta. resp. lib. 8. <lb/> +pag. 43. b. <lb/> +1. Datis tribus lateribis <lb/> +quæritur angulus D. +<lb/>[<emph style="it">tr: +Viète, Responsorum liber VIII, page 43v. <lb/> +Given three sides, there is sought angle D. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1462" xml:space="preserve"> +Analogia <lb/> +Vide syntomon 2<emph style="super">o</emph>. <lb/> +Angulus quæsitis +<lb/>[<emph style="it">tr: +Ratio <lb/> +See syntomon 2. <lb/> +Angle sought. +</emph>]<lb/> +[<emph style="it">Note: +The second case of syntomon can be found on Add MS 6782, f. 421. + </emph>]<lb/> +</s> +</p> +<pb file="add_6782_f427v" o="427v" n="855"/> +<pb file="add_6782_f428" o="428" n="856"/> +<div xml:id="echoid-div273" type="page_commentary" level="2" n="273"> +<p> +<s xml:id="echoid-s1463" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1463" xml:space="preserve"> +Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of +Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). +In the 1646 edition of Viete's <emph style="it">Opera mathematica</emph> +the triangle referred to here is to be found on page 423. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head184" xml:space="preserve" xml:lang="lat"> +Vieta. resp. lib. 8. <lb/> +pag. 43. b. <lb/> +1. Datis tribus lateribis <lb/> +quæritur angulus B. +<lb/>[<emph style="it">tr: +Viète, Responsorum liber VIII, page 43v. <lb/> +Given three sides, there is sought angle B. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1465" xml:space="preserve"> +Analogia <lb/> +numeratio in alia charta. +<lb/>[<emph style="it">tr: +Ratio <lb/> +Enumeration in the other sheet. +</emph>]<lb/> +[<emph style="it">Note: +The other sheet referred to here is probably Add MS 6782, f. 433. + </emph>]<lb/> +</s> +</p> +<pb file="add_6782_f428v" o="428v" n="857"/> +<pb file="add_6782_f429" o="429" n="858"/> +<div xml:id="echoid-div274" type="page_commentary" level="2" n="274"> +<p> +<s xml:id="echoid-s1466" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1466" xml:space="preserve"> +Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of +Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). +In the 1646 edition of Viete's <emph style="it">Opera mathematica</emph> +the triangle referred to here is to be found on page 423. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head185" xml:space="preserve" xml:lang="lat"> +Vieta. resp. lib. 8. <lb/> +pag. 43. b. +<lb/>[<emph style="it">tr: +Viète, Responsorum liber VIII, page 43v. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1468" xml:space="preserve"> +Menda in Vieta +<lb/>[<emph style="it">tr: +Wrong in Viète +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f429v" o="429v" n="859"/> +<pb file="add_6782_f430" o="430" n="860"/> +<div xml:id="echoid-div275" type="page_commentary" level="2" n="275"> +<p> +<s xml:id="echoid-s1469" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1469" xml:space="preserve"> +The reference to Fink is to <emph style="it">Geometriae rotundi libri XIIII</emph> (1583), page 364. +</s> +<lb/> +<s xml:id="echoid-s1470" xml:space="preserve"> +The reference to Regiomontanus to <emph style="it">De triangulis omnimodis libri quinque</emph> ([1464], 1533, 1561), +Book V, Proposition 1. +</s> +<lb/> +<s xml:id="echoid-s1471" xml:space="preserve"> +The reference to Viète is to the 'ALIUD' in Chapter XIX of +<emph style="it">Variorum resposorum liber VIII</emph>, Proposition 13. +See Add MS 6787, f. 223. +</s> +<lb/> +<quote xml:lang="lat"> +13 Vt rectangulum quod sit sub sinu toto & transsinuosa prima ad id quod sit +sub transsinuosa secunda & transsinuosa tertia, ita quod sit sub sinu complementi secundæ +& sinu complementi tertiæ ad id quod sit sub sinu toto & sinu complemnti primæ. +</quote> +<lb/> +<quote> +As the product of the while sine and the secant of the first to +that of the secant of the second and the secant of the third, +so is that of the sine of the complement of the second and the sine of the complement of the third +to that of the whole sine and the sine of the complement of the first. +</quote> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head186" xml:space="preserve" xml:lang="lat"> +Finkius in Geomet. rotundi. <lb/> +lib. 14, 6. pag. 364. <lb/> +Regiom. lib.5.p.1. +<lb/>[<emph style="it">tr: +Fink in Geometria rotundi, Book XIV.6, page 364. <lb/> +Regiomontanus, Book V.1. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1473" xml:space="preserve"> +Utilis propositio: <lb/> +ad indagendum angulum inclinationis <lb/> +circuli alicuius planetæ vel cometæ <lb/> +et ad alia. +<lb/>[<emph style="it">tr: +A useful proposition for delivering the angle of inclination of a circle of any planet or comet, and for other things. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1474" xml:space="preserve"> +In duobus triangulis rectangulis <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>o</mi><mi>u</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>e</mi><mi>i</mi></mstyle></math>: <lb/> +Dantur <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>o</mi></mstyle></math>. latitudo plaentæ una <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>e</mi></mstyle></math>. latitudo altera <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>e</mi></mstyle></math>. differentia longitudinum +in duobus locis. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>i</mi></mstyle></math> datur ex consqequentia <lb/> +et est arcus circuli <lb/> +planetæ. <lb/> +Quæritur angulus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +In two right-angled triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>o</mi><mi>u</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>e</mi><mi>i</mi></mstyle></math>, there are given: <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>o</mi></mstyle></math>, the latitude of one planet <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>e</mi></mstyle></math>, the latitude of the other <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>e</mi></mstyle></math>, the difference in longitude of the two locations. +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>i</mi></mstyle></math> is consequently given, and is the arc of a circle of a planet. <lb/> +There is sought angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>.</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1475" xml:space="preserve"> +Inde per Finkium <lb/> +<lb/>[...]<lb/> <lb/> +Quas Analogias deduxit ex superioribus ita: <lb/> +<lb/>[...]<lb/> <lb/> +Sed ita nullum compendium oritur, igitur inutilis commutatio. +<lb/>[<emph style="it">tr: +Thus by Fink. <lb/> +<lb/>[...]<lb/> <lb/> +Whcih ratios one deduces from the above, thus: <lb/> +<lb/>[...]<lb/> <lb/> +But in this way nothing shorter arises, therefore the change is not useful. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1476" xml:space="preserve"> +Utile compendium ita fit <lb/> +<lb/>[...]<lb/> <lb/> +Latitudo, inventa <lb/> +per syntomon. +<lb/>[<emph style="it">tr: +It may usefully be done more briefly thus: <lb/> +<lb/>[...]<lb/> <lb/> +The latitude is found by syntomon. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1477" xml:space="preserve"> +vel per 13p. vieta lib. 8. resp. pag. 37. <lb/> +<lb/>[...]<lb/> <lb/> +Hoc est:<lb/> +latitudo inventa <lb/> ++ proportione <lb/> +Hæc mutatio ergo inutilis: <lb/> +vel hæc melior quam <lb/> +illa Finkij vel originis. +<lb/>[<emph style="it">tr: +or by Proposition 13 of Viète, Responsorum liber VIII, page 37, <lb/> +<lb/>[...]<lb/> <lb/> +That is:<lb/> +the latitude found, and the proportion <lb/> +This change is therefore not useful; or this is better than that of Fink or the original. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f430v" o="430v" n="861"/> +<pb file="add_6782_f431" o="431" n="862"/> +<pb file="add_6782_f431v" o="431v" n="863"/> +<pb file="add_6782_f432" o="432" n="864"/> +<div xml:id="echoid-div276" type="page_commentary" level="2" n="276"> +<p> +<s xml:id="echoid-s1478" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1478" xml:space="preserve"> +Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of +Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). +In the 1646 edition of Viete's <emph style="it">Opera mathematica</emph> +a diagram realting to these figures is to be found on page 426. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head187" xml:space="preserve" xml:lang="lat"> +Vieta. pag. 45. <lb/> +resp. lib. 8. +<lb/>[<emph style="it">tr: +Viète, page 45, Responsorum liber VIII, page 43. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1480" xml:space="preserve"> +Anguli obliquanguli <lb/> +trianguli sphæricæ. +<lb/>[<emph style="it">tr: +Oblique-angled spherical triangles. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f432v" o="432v" n="865"/> +<pb file="add_6782_f433" o="433" n="866"/> +<div xml:id="echoid-div277" type="page_commentary" level="2" n="277"> +<p> +<s xml:id="echoid-s1481" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1481" xml:space="preserve"> +Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of +Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). +In the 1646 edition of Viete's <emph style="it">Opera mathematica</emph> +the triangles referred to here are to be found on pages 422 and 423. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head188" xml:space="preserve" xml:lang="lat"> +Vieta. rep. lib. 8 <lb/> +pag. 43. <lb/> +Anguli, rectanguli trianguli sphæricæ +<lb/>[<emph style="it">tr: +Viète, Responsorum liber VIII, page 43. Angles, in right-angled spherical triangles. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1483" xml:space="preserve"> +omnes combinationes faciunt trianguli. <lb/> +sunt quatuor tantum quia unus est 90. +<lb/>[<emph style="it">tr: +all combinatins make triangles; there are four such because one is 90. +</emph>]<lb/> +</s> +</p> +<head xml:id="echoid-head189" xml:space="preserve" xml:lang="lat"> +pag. 44. Anguli obliquianguli +<lb/>[<emph style="it">tr: +page 44. Oblique angles +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1484" xml:space="preserve"> +Nullam earum combinationium <lb/> +faciunt trainguli. <lb/> +Accipi igitur complementum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> <lb/> +ut facit Vieta. ita: +<lb/>[<emph style="it">tr: +None of these combinations makes a triangle. <lb/> +Therefore accept the complememnt of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> as Viète does, thus: +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f433v" o="433v" n="867"/> +<pb file="add_6782_f434" o="434" n="868"/> +<pb file="add_6782_f434v" o="434v" n="869"/> +<pb file="add_6782_f435" o="435" n="870"/> +<div xml:id="echoid-div278" type="page_commentary" level="2" n="278"> +<p> +<s xml:id="echoid-s1485" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1485" xml:space="preserve"> +The text referred to here is Johan Philip Lansberg, +<emph style="it">Triangulorum geometriae libri quatuor</emph> (1591). +Page 201 contains Lansberg's rule for finding a side of a spherical triangles, given its angles. +See Add MS 6787, f. 197. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head190" xml:space="preserve"> +Erallage pleuroniniche +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1487" xml:space="preserve"> +Lansberg. pag. 201. Demonstratio originis falsa est: <lb/> +Regularum aliquando +<lb/>[<emph style="it">tr: +Lansberg, page 201. The original demonstration is false for some rules. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f435v" o="435v" n="871"/> +<pb file="add_6782_f436" o="436" n="872"/> +<pb file="add_6782_f436v" o="436v" n="873"/> +<pb file="add_6782_f437" o="437" n="874"/> +<pb file="add_6782_f437v" o="437v" n="875"/> +<pb file="add_6782_f438" o="438" n="876"/> +<div xml:id="echoid-div279" type="page_commentary" level="2" n="279"> +<p> +<s xml:id="echoid-s1488" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1488" xml:space="preserve"> +A continuation of Harriot's work on the 'Dati sexti', from Chapter XIX of +Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). +Here he examines Proposition VI. +The word 'parapompe' was originally used by Viète to describe each proposition. <lb/> +</s> +<lb/> +<quote xml:lang="lat"> +VII. <lb/> +Data summa vel differentia duarum perpheriarum, quarum prosinus datam habeant rationem, +dantur singulæ <lb/> +1 Enimvero si utraque peripheria proponatur minor quadrante, vel utraque major. <lb/> +Erit, <lb/> +Vt adgregatum similium prosinuum ad differentiam eorundem, +ita sinus summæ peripheriarum ad sinuum differentiæ, +Vel ita transsinuosa complementi differentiæ ad transsinuosam complementi summæ. <lb/> +2 Quod si una e peripheriis proponatur minor quadrante, altera maior, <lb/> +Erit, <lb/> +Vt adgregatum prosinuum ad differentiam eorundem, +ita sinus differentiæ peripheriarum ad sinum adgregati, +Vel ita transsinuosa complementi summæ ad transsinuosam complementi differentiæ. +</quote> +<lb/> +<quote> +VII. Given the sum or difference of two arcs, whose tangents are in a given ratio, each is given individually. <lb/> +1. If both given arcs are less than a quadrant, or both greater, +then as the sum of the tangents is to their difference, +so is the sine of the sum of the arcs to the sine of their difference. <lb/> +Or as the secant of the complement of the difference to the secant of the complement of the sum. <lb/> +2. But if one of the given arcs is less than a quadrant, the other greater, +then as the sum of the tangents is to their difference, +so is the sine of the difference of the arcs to the sine of the sum. <lb/> +Or as the secant of the complement of the sum to the secant of the complement of the difference. +</quote> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head191" xml:space="preserve" xml:lang="lat"> +Vieta. 37.b. lib. 8. resp. <lb/> +<foreign xml:lang="gre">parapompe</foreign> <lb/> +Dati Septimi. <lb/> +Data summa vel differentia duarum peripheriarum,  <lb/> +et ratione O. +<lb/>[<emph style="it">tr: +Viète, page 37v, Responsorum liber VIII. <lb/> +Parapompe <lb/> +Seventh proposition. <lb/> +Given the sum or difference of two arcs and the ratio of their tangents. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1490" xml:space="preserve"> +1. <lb/> +utraque minor <lb/> +quadrante +<lb/>[<emph style="it">tr: +1. both less than a quadrant +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1491" xml:space="preserve"> +Interpetatio <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, una peripheria, minor quadrante. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, altera peripheria minor quadrante. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi><mo>=</mo><mi>b</mi><mi>a</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>c</mi></mstyle></math>, differentia peripheriæ. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi></mstyle></math>, tangens <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math>, tangens <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>f</mi></mstyle></math>, differentia tangentium. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi></mstyle></math>, sinus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>l</mi></mstyle></math>, sinus differentiæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Interpetation <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, one arc, less than a quadrant. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, the other arc, less than a quadrant. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi><mo>=</mo><mi>b</mi><mi>a</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>c</mi></mstyle></math>, the difference of the arcs. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi></mstyle></math>, tangent to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math>, tangento <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>f</mi></mstyle></math>, the difference of the tangents. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi></mstyle></math>, sine of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>l</mi></mstyle></math>, sine of the difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1492" xml:space="preserve"> +2. <lb/> +una maior +<lb/>[<emph style="it">tr: +2. one greater +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1493" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> peripheria maior quadrante. <lb/> +cætera ut supra +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> is an arc greater than a quadrant; <lb/> +the rest is as above. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f438v" o="438v" n="877"/> +<pb file="add_6782_f439" o="439" n="878"/> +<div xml:id="echoid-div280" type="page_commentary" level="2" n="280"> +<p> +<s xml:id="echoid-s1494" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1494" xml:space="preserve"> +A continuation of Harriot's work on the 'Dati sexti', from Chapter XIX of +Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). +Here he examines Proposition VI. +</s> +<lb/> +<quote xml:lang="lat"> +VI. <lb/> +Data summa vel differentia duarum perpheriarum, quarum sinus datam habeant rationem, +dantur singulares peripheriæ. <lb/> +1 Enimvero si utraque peripheria proponitur minor quadrante, vel utraque major. <lb/> +Erit, <lb/> +Vt adgregatum similium sinuum ad differentiam eorundem, +ita prosinus dimidiæ summæ peripheriarum ad prosinum dimidiæ differentiæ earundem, +Vel ita prosinus complementi dimidiæ differentiæ peripheriarum ad prosinum complementi dimidiæ summæ. <lb/> +2 Quod si una e peripheriis proponatur minor quadrante, altera maior, <lb/> +Erit, <lb/> +Vt adgregatum sinuum ad differentiam eorundem, +ita prosinus dimidiæ differentiæ peripheriarum ad prosinum dimidiæ summæ, +Vel ita prosinus complementi dimidiæ summæ ad prosinum complementi dimidiæ differentiæ. +</quote> +<lb/> +<quote> +VI. Given the sum or difference of two arcs, whose sines are in a given ratio, each arc is given individually. <lb/> +1. If both given arcs are less than a quadrant of the circle, or both greater, +then as the sum of those sines is to their difference, +so is the tangent of half the sum of the arcs to the tangent of half their difference. <lb/> +Or as the tangent of the complement of half the difference of the arcs to +the tangent of the complement of half the sum. <lb/> +2. But if one of the given arcs is less than a quadrant, the other greater, +then as the sum of the sines is to their difference, +so is the tangent of half the difference of the arcs to the tangent of half their sum. <lb/> +Or as the tangent of the complement of half the sum to the tangent of the complement of half the difference. +</quote> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head192" xml:space="preserve" xml:lang="lat"> +Vieta. lib. 8. resp. <lb/> +pag. 37. <lb/> +VI. <lb/> +Data summa vel differentia <lb/> +et ratione <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Î¥</mo></mstyle></math>. +<lb/>[<emph style="it">tr: +Viète, Responsorum liber VIII, page 37, VI. <lb/> +Given the sum or difference and the ratio of their sines. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1496" xml:space="preserve"> +Interpetatio <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, una peripheria minor quadrante. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, altera peripheria minor quadrante. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>g</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi></mstyle></math> ratio sinuum <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>=</mo><mi>m</mi><mi>c</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>m</mi></mstyle></math> differentia inter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>h</mi></mstyle></math> differentia inter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>g</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Interpetation <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> one arc, less than a quadrant <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> the other arc, less than a quadrant <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>g</mi><mo>:</mo><mi>g</mi><mi>c</mi></mstyle></math> ratio of the sines <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>=</mo><mi>m</mi><mi>c</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>m</mi></mstyle></math> difference between <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>h</mi></mstyle></math> difference between <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>g</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1497" xml:space="preserve"> +Hic <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> est maior quadrante. non tamen variat casum. <lb/> +Menda igitur in Vieta +<lb/>[<emph style="it">tr: +Here <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> is greater than a quadrant, nevertheless, the case does not change. <lb/> +Therefore wrong in Viète. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f439v" o="439v" n="879"/> +<pb file="add_6782_f440" o="440" n="880"/> +<div xml:id="echoid-div281" type="page_commentary" level="2" n="281"> +<p> +<s xml:id="echoid-s1498" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1498" xml:space="preserve"> +A continuation of Harriot's work on the 'Dati sexti', from Chapter XIX of +Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). +Here he examines Propositions V.1 and V.2. +</s> +<lb/> +<quote xml:lang="lat"> +V. <lb/> +Data differentia duarum perpheriarum, quarum sinus datam habeant rationem, dantur singulæ <lb/> +1 Enimvero si differentia sit maior quadrante circuli. <lb/> +Erit, <lb/> +Vt sinus componentium primæ ad sinum secundæ, +ita transsinuosa complementi differentiæ ad prosinum complementi primæ minus prosinu complementi differentiæ. <lb/> +Cum autem prima sumetur maior quadrante, secunda sumetur minor, & contra. <lb/> +2 Et si differentia minor quadrante circuli, differentes autem peripheriæ diversæ sint speciei, <lb/> +Erit, <lb/> +Vt sinus primæ ad sinum secundæ, +ita transsinuosa complementi differentiæ ad prosinum complementi differentiæ, plus prosinu complementi primæ. <lb/> +Cum autem prima sumetur maior quadrante, secunda sumetur minor, & contra. +</quote> +<lb/> +<quote> +V. Given the difference of two arcs, whose sines are in a given ratio, each is given individually. <lb/> +1. If the difference is greater than a quadrant of the circle, +then as the sine of the first component is to the sine of the second, +so is the secant of the complement of the difference to the tangent of the complement of the first +minus the tangent of the complement of the difference. <lb/> +Moreover, when the first it taken greater than a quadrant, the second is taken less, and conversely. <lb/> +2. And if the difference is greater than a quadrant of the circle, but the different arcs have different signs, +then as the sine of the first component is to the sine of the second, +so is the secant of the complement of the difference to the tangent of the complement of the difference +plus the tangent of the complement of the first. +</quote> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head193" xml:space="preserve" xml:lang="lat"> +Vieta. lib. 8. resp. <lb/> +pag. 37. <lb/> +V. <lb/> +Data differentia duarum perpheriarum <lb/> +et ratione <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Î¥</mo></mstyle></math>. +<lb/>[<emph style="it">tr: +Viète, Responsorum liber VIII, page 37, V. <lb/> +Given the difference of two arcs and the ratio of their sines. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1500" xml:space="preserve"> +1. <lb/> +differentia <lb/> +maior quad. +<lb/>[<emph style="it">tr: +1. the difference greater than a quadrant +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1501" xml:space="preserve"> +Interpetatio <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> una peripheria. cui æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> altera <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi></mstyle></math> differentia <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>h</mi></mstyle></math> angulus rectus <lb/> +<lb/>[...]<lb/> +<lb/>[<emph style="it">tr: +Interpetation <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> one arc, to which <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi></mstyle></math> is equal <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> the other <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi></mstyle></math> the difference +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>h</mi></mstyle></math>, a right angle <lb/> +<lb/>[...]<lb/> +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1502" xml:space="preserve"> +Ut minor terminus sit primum proportionalium. <lb/> +Fiat angulus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>e</mi><mi>r</mi></mstyle></math>, æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>k</mi></mstyle></math> angulo, qui est angulus complementi differentiæ. <lb/> +Tum triangula <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>r</mi><mi>g</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi><mi>k</mi><mi>e</mi></mstyle></math> sunt æquiangula. nam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi><mi>k</mi><mi>e</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>r</mi><mi>g</mi></mstyle></math> sunt anguli <lb/> +residui æqualia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>r</mi><mi>e</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>k</mi><mi>e</mi></mstyle></math>. et anguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>e</mi><mi>l</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>g</mi><mi>e</mi></mstyle></math> sunt æquales ab paralleles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>q</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>l</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +If the smaller term is the first proportional. <lb/> +Construct anlge <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>e</mi><mi>r</mi></mstyle></math> equal to angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>k</mi></mstyle></math>, which is the angle of the complement of the difference. <lb/> +Then triangles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>r</mi><mi>g</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi><mi>k</mi><mi>e</mi></mstyle></math> are equiangular, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi><mi>k</mi><mi>e</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>r</mi><mi>g</mi></mstyle></math> are residual angles from +the equal angles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>r</mi><mi>e</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>k</mi><mi>e</mi></mstyle></math>; and angles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>e</mi><mi>l</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>g</mi><mi>e</mi></mstyle></math> are equals by the parallels <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>q</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>l</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1503" xml:space="preserve"> +2. <lb/> +differentia <lb/> +minor quad: <lb/> +peripheria <lb/> +una minor, <lb/> +altera maior <lb/> +quadrante. +<lb/>[<emph style="it">tr: +2. the difference less than a quadrant; one arc less tha, the other greater than a quadrant. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1504" xml:space="preserve"> +Superiora verba et litteræ <lb/> +deservierunt etiam hinc diagram-<lb/> +mati; et concludant: +<lb/>[<emph style="it">tr: +The above words and letters serve also for this diagram; and end with: +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1505" xml:space="preserve"> +Ut minor terminus sit primus proportionalium. <lb/> +Hic anguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>r</mi><mi>g</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>k</mi><mi>l</mi></mstyle></math>, sunt complemmentat <lb/> +æqualium angulum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>e</mi><mi>n</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>e</mi><mi>h</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +As the lesser term is the first proportional. <lb/> +Here angles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>r</mi><mi>g</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>k</mi><mi>l</mi></mstyle></math> are complements of equal angles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>e</mi><mi>n</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>e</mi><mi>h</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f440v" o="440v" n="881"/> +<pb file="add_6782_f441" o="441" n="882"/> +<div xml:id="echoid-div282" type="page_commentary" level="2" n="282"> +<p> +<s xml:id="echoid-s1506" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1506" xml:space="preserve"> +A continuation of Harriot's work on the 'Dati sexti', from Chapter XIX of +Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). +Here he examines Proposition V.3. +</s> +<lb/> +<quote xml:lang="lat"> +V. <lb/> +Data differentia duarum perpheriarum, quarum sinus datam habeant rationem, dantur singulæ <lb/> +<lb/>[...]<lb/> <lb/> +3 Et si denique differentia sit minor quadrante, +utraque vero differentium vel quadrante minor vel utraque quadrante maior, +ac prima quidem intelligatur ea cui debetur sinus major, secunda cui minor, <lb/> +Erit, <lb/> +Vt sinus primæ ad sinum secundæ, +ita transsinuosa complementi differentiæ ad prosinum complementi differentiæ minus prosinu complementi primæ. <lb/> +Et, <lb/> +Vt sinus primæ ad sinum secundæ, +ita transsinuosa complementi differentiæ ad prosinum complementi differentiæ plus prosinu complementi primæ. +</quote> +<lb/> +<quote> +V. Given the difference of two arcs, whose sines are in a given ratio, each is given individually. <lb/> +<lb/>[...]<lb/> <lb/> +3. And if finally the difference is less than a quadrant, +then as the sine of the first component is to the sine of the second, +so is the secant of the complement of the difference to the tangent of the complement of the difference +minus the tangent of the complement of the first. +And as the sine of the first is to the sine of the second, +so is the secant of the complement of the difference to the tangent of the complement of the difference +plus the tangent of the complement of the first. <lb/> +</quote> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head194" xml:space="preserve" xml:lang="lat"> +Vieta. lib. 8. resp. pag. 37. <lb/> +V. <lb/> +Data differentia duarum perpheriarum <lb/> +et ratione <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Î¥</mo></mstyle></math>. +<lb/>[<emph style="it">tr: +Viète, Responsorum liber VIII, page 37, V. <lb/> +Given the difference of two arcs and the ratio of their sines. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1508" xml:space="preserve"> +3. <lb/> +differentia <lb/> +minor quad. <lb/> +et <lb/> +utraque <lb/> +peripheriæ. +<lb/>[<emph style="it">tr: +3. the difference greater than a quadrant, and both the arcs. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1509" xml:space="preserve"> +Verba <emph style="super">et litteræ</emph> superiores <lb/> +diagrammatis <lb/> +2, et 1, deservierunt <lb/> +etiam huic. +<lb/>[<emph style="it">tr: +The words and letters for the above diagrams, 2 and 1, serve also for this, +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1510" xml:space="preserve"> +Ut minor terminus sit primum proportinalium. +<lb/>[<emph style="it">tr: +If the smaller term is the first proportional. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f441v" o="441v" n="883"/> +<pb file="add_6782_f442" o="442" n="884"/> +<div xml:id="echoid-div283" type="page_commentary" level="2" n="283"> +<p> +<s xml:id="echoid-s1511" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1511" xml:space="preserve"> +A continuation of Harriot's work on the 'Dati sexti', from Chapter XIX of +Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). +Here he examines Proposition IV. +</s> +<lb/> +<quote xml:lang="lat"> +IV. <lb/> +Data peripheria composita e duabus peripheriis, quarum sinus datam habeant rationem, dantur singulæ. <lb/> +1 Enimvero si composita minor est circuli quadrante. <lb/> +Erit, <lb/> +Vt sinus componentium primæ ad sinum secundæ, +ita transsinuosa complementi compositæ ad prosinum complementi primæ minus prosinu complementi compositæ. <lb/> +2 Et si composita maior est quadrante, utraque vero componentium minor quadrante. <lb/> +Erit, <lb/> +Vt sinus primæ ad sinum secundæ, +ita s complementi compositæ ad prosinum complementi compositæ plus prosinu complementi primæ. <lb/> +3 Et si denique componentium peripheriarum primæ sit minor quadrante, secunda maior, <lb/> +Erit, <lb/> +Vt sinus primæ ad sinum secundæ, +ita transsinuosa complementi compositæ ad prosinum complementi compositæ minus prosinu complementi primæ. <lb/> +Et, <lb/> +Vt sinus primæ ad sinum secundæ, +ita transsinuosa complementi compositæ ad prosinum complementi compositæ plus prosinu secundæ. <lb/> +</quote> +<lb/> +<quote> +IV. Given the sum of two arcs, whose sines are in a given ratio, each is given individually. <lb/> +1. If the sum is less than a quadrant of the circle, then as the sine of the first component is to the sine of the second, +so is the secant of the complement of the sum to the tangent of the complement of the first +minus the tangent of the complement of the sum. <lb/> +2. And if the sum is greater than a quadrant, but both components are less than a quadrant, +then as the sine of the first component is to the sine of the second, +so is the secant of the complement of the sum to the tangent of the complement of the sum +plus the tangent of the complement of the first. <lb/> +3. And if finally the first component of the sum is less than a quadrant, the second greater, +then as the sine of the first component is to the sine of the second, +so is the secant of the complement of the sum to the tangent of the complement of the sum +minus the tangent of the complement of the first. +And as the sine of the first is to the sine of the second, +so is the secant of the complement of the sum to the tangent of the complement of the sum +plus the tangent of the second. +</quote> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head195" xml:space="preserve" xml:lang="lat"> +Vieta. lib. 8. resp. <lb/> +pag. 38. b. <lb/> +IIII. <lb/> +Data peripheria composita <lb/> +et ratione <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Î¥</mo></mstyle></math>. +<lb/>[<emph style="it">tr: +Viète, Responsorum liber VIII, page 38v, IV. <lb/> +Given a sum of arcs and the ratio of their sines, 2. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1513" xml:space="preserve"> +1. <lb/> +composita minor <lb/> +quadrante +<lb/>[<emph style="it">tr: +1. the sum less than a quadrant +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1514" xml:space="preserve"> +Interpetatio <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> una peripheria <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> altera peripheria <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math>,<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math>: ratio sinuum <lb/> +<lb/>[...]<lb/> +<lb/>[<emph style="it">tr: +Interpetation <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> one arc <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> the other <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi><mo>:</mo><mi>d</mi><mi>c</mi></mstyle></math>, the ratio of sines <lb/> +<lb/>[...]<lb/> +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1515" xml:space="preserve"> +2. <lb/> +composita maior; <lb/> +utraque minor +<lb/>[<emph style="it">tr: +2. the sum greater; both [arcs] less +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1516" xml:space="preserve"> +3. <lb/> +composita <lb/> +maior: <lb/> +una minor, <lb/> +altera maior +<lb/>[<emph style="it">tr: +3. the sum greater; one [arc] less, the other greater +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1517" xml:space="preserve"> +Inde cum 2. <lb/> +Menda in Vieta +<lb/>[<emph style="it">tr: +Hence like 2, wrong in Viète. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1518" xml:space="preserve"> +Aliter pro 3. Ut maior terminus sit primus proportionalium. <lb/> +<lb/>[<emph style="it">tr: +Another way for 3, when the greater term is the first proportional. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f442v" o="442v" n="885"/> +<pb file="add_6782_f443" o="443" n="886"/> +<div xml:id="echoid-div284" type="page_commentary" level="2" n="284"> +<p> +<s xml:id="echoid-s1519" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1519" xml:space="preserve"> +A continuation of Harriot's work on the 'Dati sexti', from Chapter XIX of +Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). +Here he examines Proposition III.2. +</s> +<lb/> +<quote xml:lang="lat"> +III. <lb/> +Data summa vel differentia duarum peripheriarum, quarum transsinuosae datam habeant rationem, dantur singulæ. <lb/> +<lb/>[...]<lb/> <lb/> +2 Quod si une e peripheriis proponitur minor quadrante, altera maior <lb/> +Erit, <lb/> +Vt adgregatum similium transsinuousuarum ad differentiam earundem, +ita prosinus dimidia differentiæ peripheriarum ad prosinum complementi dimidæ summæ, +Et ita prosinus dimidiæ summæ ad prosinum complementi dimidiæ differentiæ. +</quote> +<lb/> +<quote> +III. Given the sum or difference of two arcs, whose secants are in a given ratio, each is given individually. <lb/> +2. But if one of the arcs is less than a quadrant, the ohter greater, then as the sum of the similar secants is +to their difference, so is the tangent of half the difference of the arcs +to the tangent of the complement of half the sum. +And so is the tangent of half the sum to the tangent of the complement of half the difference. +</quote> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head196" xml:space="preserve" xml:lang="lat"> +Vieta. lib. 8. resp. <lb/> +pag. 38. b. <lb/> +III. <lb/> +Data summa vel differentia <lb/> +et ratione <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math>. 2. +<lb/>[<emph style="it">tr: +Viète, Responsorum liber VIII, page 38v, III. <lb/> +Given the sum or difference of two arcs and the ratio of their secants, 2. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1521" xml:space="preserve"> +2. <lb/> +peripheria <lb/> +una minor <lb/> +quadrante; <lb/> +altera maior. +<lb/>[<emph style="it">tr: +2. one arc is less than a quadrant, the other greater +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1522" xml:space="preserve"> +Interpetatio <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> una peripheria, minor quadrante <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> altera, maior quadrante <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>i</mi><mo>-</mo><mi>a</mi><mi>b</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>c</mi></mstyle></math>, differentia inter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>c</mi></mstyle></math>, dimidia differentia <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>g</mi></mstyle></math>, eius tangens <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>g</mi><mo>=</mo><mi>k</mi><mi>g</mi></mstyle></math>, et parallelæ <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi></mstyle></math> secans peripheriæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi></mstyle></math>, secans peripheriæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>t</mi><mo>=</mo><mi>e</mi><mi>d</mi><mo>=</mo><mi>e</mi><mi>f</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi><mi>h</mi></mstyle></math>, differentia secantium <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi></mstyle></math> <lb/> +fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>y</mi><mo>=</mo><mi>e</mi><mi>t</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>i</mi><mi>k</mi></mstyle></math> dimidia summa compositæ <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>m</mi></mstyle></math>, complementum dimidiæ summæ <lb/> +<lb/>[...]<lb/> +<lb/>[<emph style="it">tr: +Interpetation <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> one arc, less than a quadrant <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> the other, greater than a quadrant <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>i</mi><mo>-</mo><mi>a</mi><mi>b</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>c</mi></mstyle></math>, the difference between <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>c</mi></mstyle></math>, half the difference <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>g</mi></mstyle></math>, its tangent <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>g</mi><mo>=</mo><mi>k</mi><mi>g</mi></mstyle></math>, and parallels <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi></mstyle></math> secant of the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi></mstyle></math>, secant of the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>t</mi><mo>=</mo><mi>e</mi><mi>d</mi><mo>=</mo><mi>e</mi><mi>f</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi><mi>h</mi></mstyle></math>, difference of the secants <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi></mstyle></math> <lb/> +fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>y</mi><mo>=</mo><mi>e</mi><mi>t</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>i</mi><mi>k</mi></mstyle></math> half the sum of the composite arc <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>m</mi></mstyle></math>, complement of half the sum <lb/> +<lb/>[...]<lb/> +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f443v" o="443v" n="887"/> +<pb file="add_6782_f444" o="444" n="888"/> +<div xml:id="echoid-div285" type="page_commentary" level="2" n="285"> +<p> +<s xml:id="echoid-s1523" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1523" xml:space="preserve"> +A continuation of Harriot's work on the 'Dati sexti', from Chapter XIX of +Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). +Here he examines Proposition III.1. +The word 'parapompe' was originally used by Viète to describe each proposition. <lb/> +The page number 36v given in the top left hand corner is incorrect; it should be 38v. +</s> +<lb/> +<quote xml:lang="lat"> +III. <lb/> +Data summa vel differentia duarum peripheriarum, quarum transsinuosae datam habeant rationem, dantur singulæ. <lb/> +1 Enimvero si utraque peripheria proponatur minor quadrante vel utraque major. <lb/> +Erit, <lb/> +Vt adgregatum similium transsinuousuarum ad differentiam earundem, +ita prosinus complementi dimidia summæ peripheriæ ad prosinum dimidæ differentiæ, +Et ita prosinus complementi dimidiæ differentiæ ad prosinum dimidiæ summæ. +</quote> +<lb/> +<quote> +III. Given the sum or difference of two arcs, whose secants are in a given ratio, each is given individually. <lb/> +1. If both the given arcs are less than a quadrant or both greater, then as the sum of the similar secants is +to their difference, so is the tangent of the complement of half the sum of the arcs +to the tangent of half the difference. +And so is the tangent of the complement of half the difference to the tangent of half the sum. +</quote> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head197" xml:space="preserve" xml:lang="lat"> +Vieta. lib. 8. resp. <lb/> +pag. 36. b. <lb/> +III. <lb/> +et pag. 38. <lb/> +1. <lb/> +data peripheria <lb/> +composita. <lb/> +et ratione <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Ψ</mo></mstyle></math>. <lb/> +<foreign xml:lang="gre">parapompe</foreign>. 3. Data summa vel differentia duarum peripheriarum <lb/> +et ratione <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math>. 1. +<lb/>[<emph style="it">tr: +Viète, Responsorum liber VIII, page 36v, III. <lb/> +and page 38, 1, given the sum of arcs and the ratio of their secants <lb/> +Parapompe III: Given the sum or difference of two arcs and the ratio of their secants. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1525" xml:space="preserve"> +I.1. <lb/> +peripheria utraque <lb/> +minor quadrante +<lb/>[<emph style="it">tr: +I.1.either arc is less than a quadrant +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1526" xml:space="preserve"> +Arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. Tangens <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi></mstyle></math>. <lb/> +Arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. Tangens <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>=</mo><mi>b</mi><mi>i</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>c</mi><mo>,</mo><mi>d</mi><mi>i</mi><mi>f</mi><mi>f</mi><mi>e</mi><mi>r</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>a</mi><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi></mstyle></math> ab <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>t</mi></mstyle></math> bc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>.</mo></mstyle></math><lb/> +Aggregatum tangentium <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi><mo>+</mo><mi>e</mi><mi>h</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>h</mi></mstyle></math>, est differentia tangentium <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>k</mi></mstyle></math> est dimidium arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>c</mi></mstyle></math>. <lb/> +Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>r</mi></mstyle></math> est dimidium totius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>s</mi></mstyle></math> est complementum arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>r</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>n</mi></mstyle></math> est tangens complementi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>k</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>l</mi></mstyle></math> est tangens arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>k</mi></mstyle></math>, secans <lb/> +dimidij differentiæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>c</mi></mstyle></math>. <lb/> +Lineæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>n</mi></mstyle></math> sit parallellæ +<lb/>[<emph style="it">tr: +Arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, tangent <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi></mstyle></math>. <lb/> +Arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, tangens <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>=</mo><mi>b</mi><mi>i</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>c</mi><mo>,</mo><mi>t</mi><mi>h</mi><mi>e</mi><mi>d</mi><mi>i</mi><mi>f</mi><mi>f</mi><mi>e</mi><mi>r</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>b</mi><mi>e</mi><mi>t</mi><mi>w</mi><mi>e</mi><mi>e</mi><mi>n</mi></mstyle></math> ab <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>n</mi><mi>d</mi></mstyle></math> bc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>.</mo></mstyle></math><lb/> +Sum of the tangents <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi><mo>+</mo><mi>e</mi><mi>h</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>h</mi></mstyle></math> is the difference between the tangents <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>k</mi></mstyle></math> is half the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>c</mi></mstyle></math>. <lb/> +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>r</mi></mstyle></math> is half the total <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>s</mi></mstyle></math> is the complement of the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>r</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>n</mi></mstyle></math> is the tangent of the complement <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>k</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>l</mi></mstyle></math> is the tangent of the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>k</mi></mstyle></math>, cutting half the difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>c</mi></mstyle></math>. <lb/> +The lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>n</mi></mstyle></math> are parallel. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1527" xml:space="preserve"> +<foreign xml:lang="gre">parapompe</foreign> pro pag. 38. Data peripheria <lb/> +Triangles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>d</mi><mi>e</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>x</mi><mi>u</mi></mstyle></math> are equiangular <lb/> +For angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>u</mi><mi>x</mi></mstyle></math> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>d</mi></mstyle></math>, because either is the complement of angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>e</mi><mi>u</mi></mstyle></math>, +and it is obvious that angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi><mi>h</mi></mstyle></math> is equal to angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>x</mi><mi>e</mi></mstyle></math>; +therefore a third of angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>u</mi><mi>x</mi></mstyle></math> is equal to angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi><mi>d</mi></mstyle></math>. <lb/> +<lb/>[...]<lb/> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>a</mi></mstyle></math>, is the tangent of the conplement of the sum. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>x</mi></mstyle></math>, is the tangent of the second. +<lb/>[<emph style="it">tr: +<foreign xml:lang="gre">parapompe</foreign> for page 38. Given the sum of the arcs. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>=</mo><mi>b</mi><mi>i</mi></mstyle></math> <lb/> +Triangula <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>d</mi><mi>e</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>x</mi><mi>u</mi></mstyle></math> sunt æquiangula <lb/> +Nam angulus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>u</mi><mi>x</mi></mstyle></math> est æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>d</mi></mstyle></math> quia <lb/> +uterque est complementum anguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>e</mi><mi>u</mi></mstyle></math> <lb/> +Et manifestum est angulus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi><mi>h</mi></mstyle></math> esse <lb/> +æqualem angulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>x</mi><mi>e</mi></mstyle></math>; Ergo tertius <lb/> +angulus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>u</mi><mi>x</mi></mstyle></math> æqualis est angulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi><mi>d</mi></mstyle></math>. <lb/> +<lb/>[...]<lb/> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>a</mi></mstyle></math>, est prosinus complementi compositæ <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>x</mi></mstyle></math>, est prosinus secundæ. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f444v" o="444v" n="889"/> +<pb file="add_6782_f445" o="445" n="890"/> +<div xml:id="echoid-div286" type="page_commentary" level="2" n="286"> +<p> +<s xml:id="echoid-s1528" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1528" xml:space="preserve"> +At the end of Chapter XIX of <emph style="it">Variorum responsorum liber VIII</emph> (1593), +under the heading 'DATI SEXTI', Viète listed six propositions for finding either of two arcs +given the sums of differences of the arcs and the ratio of their secants. +In the 1646 edition of Viète's <emph style="it">Opera mathematica</emph> +the six propositions are to be found on page 413. +</s> +<lb/> +<s xml:id="echoid-s1529" xml:space="preserve"> +On this page Harriot examines Proposition I, part 2. +</s> +<lb/> +<quote xml:lang="lat"> +I. <lb/> +Data peripheria composita e duabus peripheriis, quarum transsinuosae datam habeant rationem, dantur singulæ. <lb/> +<lb/>[...]<lb/> <lb/> +2 Et si composita major est quadrante circuli, utraque vero componentium minor quadrante, <lb/> +Erit, <lb/> +Vt transsinuosa primæ ad transsinuousa secundæ, ita transsinuosa complementi composita ad prosinum secunda +minus prosinus complementa compositæ. +</quote> +<lb/> +<quote> +I.Given an arc composed of two others, whose secants are in a given ratio, each is known individually. <lb/> +<lb/>[...]<lb/> <lb/> +2. And if the sum is greater than a quarter of a circle, but each component is less than a quarter, then +as the secant of the first is to the secant of the second, so is the secant of the complement of the sum +to the tangent of the second minus the tangent of the complement of the sum. +</quote> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head198" xml:space="preserve" xml:lang="lat"> +Vieta. lib. 8. resp. <lb/> +pag. 38. <lb/> +Data peripheria composita duarum perpheriarum, <lb/> +et ratione <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ψ</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Viète, Responsorum liber VIII, page 38. <lb/> +Given an arc composed of two arcs, and the ratio of their secants. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1531" xml:space="preserve"> +composita <lb/> +maior <lb/> +utrusque minor +<lb/>[<emph style="it">tr: +the sum is greater or lesser than +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1532" xml:space="preserve"> +triangula <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>d</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi><mi>g</mi></mstyle></math> sunt equiangula. nam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>e</mi><mi>g</mi></mstyle></math> æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi><mi>d</mi></mstyle></math>, quia uterque <lb/> +est complementum anguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>g</mi></mstyle></math>. et anguli ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> sunt æquales propter similia <lb/> +trianguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi><mi>i</mi></mstyle></math>, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi><mi>i</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Triangles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>d</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi><mi>g</mi></mstyle></math> are equiangular. +For <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>e</mi><mi>g</mi></mstyle></math> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi><mi>d</mi></mstyle></math>, because either is the complement of angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>g</mi></mstyle></math>; +and the angles at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> and 4 d <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>r</mi><mi>e</mi><mi>e</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>l</mi><mi>b</mi><mi>e</mi><mi>c</mi><mi>a</mi><mi>u</mi><mi>s</mi><mi>e</mi><mi>o</mi><mi>f</mi><mi>s</mi><mi>i</mi><mi>m</mi><mi>i</mi><mi>l</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>a</mi><mi>n</mi><mi>g</mi><mi>l</mi><mi>e</mi><mi>s</mi></mstyle></math> adi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>n</mi><mi>d</mi></mstyle></math> bgi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>.</mo></mstyle></math></emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1533" xml:space="preserve"> +Non refert an <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> sit <lb/> +minor vel maior quam <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +It does not matter whether <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> is less than or greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f445v" o="445v" n="891"/> +<pb file="add_6782_f446" o="446" n="892"/> +<div xml:id="echoid-div287" type="page_commentary" level="2" n="287"> +<p> +<s xml:id="echoid-s1534" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1534" xml:space="preserve"> +A continuation from Add MS 6782, f. 445, of Harriot's work on the 'Dati sexti' from Chapter XIX of Viète's +<emph style="it">Variorum responsorum liber VIII</emph> (1593). +Here he examines Proposition I, part 3, which he has divided as 3 and 4, +according to whether one takes the minus or plus sign in the final sentence of the statement. +</s> +<lb/> +<quote xml:lang="lat"> +I. <lb/> +Data peripheria composita e duabus peripheriis, quarum transsinuosae datam habeant rationem, dantur singulæ. <lb/> +<lb/>[...]<lb/> <lb/> +3 Et si denique componentium peripheriarum prima fit minor quadrante, secunda major, <lb/> +Erit, <lb/> +Vt transsinuosa primæ ad transsinuosam secundæ, ita transsinuosa complementi compositæ +ad prosinum complementi compositæ minus prosinu secundæ. +Et, <lb/> +Vt transsinuosa secunda ad transsinuousam primæ, ita transsinuosa complementi compositæ +ad prosinum complementi compositæ plus prosinu secundæ. +</quote> +<lb/> +<quote> +I.Given an arc composed of two others, whose secants are in a given ratio, each is known individually. <lb/> +<lb/>[...]<lb/> <lb/> +3. And if further the first arc is less than a quadrant, the second greater, then +as the secant of the first is to the secant of the second, so is the secant of the complement of the sum +to the sine of the complement of the sum minus the sine of the second, <lb/> +and <lb/> +[4.] as the secant of the second to the secant of the first, so is the secant of the complement of the sum +to the sine of the complement of the sum plus the sine of the second. +</quote> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head199" xml:space="preserve" xml:lang="lat"> +Vieta. resp. lib. 8. <lb/> +pag. 38. <lb/> +3. et 4. <lb/> +Data peripheria composita e duabus peripheriis, quarum transsinuosæ <emph style="super"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Ψ</mo></mstyle></math></emph> +datam habeant rationem, dantur singulæ. +<lb/>[<emph style="it">tr: +Viète, Responsorum liber VIII, page 38, Propositions 3 and 4. <lb/> +Given an arc composed of two arcs, whose secants are in a given ratio, each is given separately. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1536" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math> peripheria composita <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> prima minor quadrante <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> secunda maior quadrante <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi></mstyle></math> Transsinuosa <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, primæ <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>c</mi></mstyle></math> Transsinuosa <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, secundæ +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math> is the sum of the arcs <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> is the first, less than a quadrant <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> is the second, greater than a quadrant <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi></mstyle></math> is the secant of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, the first <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>c</mi></mstyle></math> is the secant of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, the second +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1537" xml:space="preserve"> +1<emph style="super">o</emph>. Triangula <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi><mi>g</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi><mi>o</mi></mstyle></math> sunt æquiangula; +propter parallelas <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>o</mi></mstyle></math>. <lb/> +2<emph style="super">o</emph>. Triangula <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>d</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi><mi>m</mi></mstyle></math> sunt æquiangula. +nam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi><mi>d</mi></mstyle></math> est complementum anguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>h</mi></mstyle></math> inde æqualis <lb/> +angulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>k</mi></mstyle></math>, cui æqualis fit per constructionem <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>e</mi><mi>m</mi></mstyle></math>. anguli igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>e</mi><mi>m</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi><mi>d</mi></mstyle></math> sunt æquales <lb/> +et anguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>d</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi><mi>m</mi></mstyle></math> sunt æquales propter parallelas <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>f</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi></mstyle></math>. Tertius ergo angulus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi><mi>g</mi></mstyle></math> <lb/> +est æqualis tertio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>m</mi><mi>e</mi></mstyle></math>. <lb/> +3<emph style="super">o</emph>. Arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>k</mi></mstyle></math> sunt æquales. nam arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> est arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi><mo>-</mo><mi>b</mi><mi>k</mi></mstyle></math>, +hoc est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi><mo>-</mo><mi>k</mi><mi>n</mi></mstyle></math>. <lb/> +4<emph style="super">o</emph>. Triangula <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>h</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>e</mi><mi>o</mi></mstyle></math> sunt æqualia et æquiangula. +<lb/>[<emph style="it">tr: +1. Triangles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi><mi>g</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi><mi>o</mi></mstyle></math> are equiangular, because <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>o</mi></mstyle></math> are parallel. <lb/> +2. Triangles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>d</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi><mi>m</mi></mstyle></math> are equiangular, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi><mi>d</mi></mstyle></math> is the complement of angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>h</mi></mstyle></math>, +thence equal to angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>k</mi></mstyle></math>, to which <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>e</mi><mi>m</mi></mstyle></math> is equal by construction; +therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>e</mi><mi>m</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi><mi>d</mi></mstyle></math> are equal, +and angles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>d</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi><mi>m</mi></mstyle></math> are equal because <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>f</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>h</mi></mstyle></math> are parallel. +Therefore a third of angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi><mi>g</mi></mstyle></math> is equal to a third of angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>m</mi><mi>e</mi></mstyle></math>. <lb/> +3. Arcs <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>k</mi></mstyle></math> are equal, for arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> is arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi><mo>-</mo><mi>b</mi><mi>k</mi></mstyle></math>, +that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi><mo>-</mo><mi>k</mi><mi>n</mi></mstyle></math>. <lb/> +4. Triangles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>h</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>e</mi><mi>o</mi></mstyle></math> are equal and equiangular.<lb/> +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1538" xml:space="preserve"> +peripheria <lb/> +una minor <lb/> +altera maior <lb/> +quadrante +<lb/>[<emph style="it">tr: +one of the arcs is less than, the other greater than a quadrant +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f446v" o="446v" n="893"/> +<pb file="add_6782_f447" o="447" n="894"/> +<div xml:id="echoid-div288" type="page_commentary" level="2" n="288"> +<p> +<s xml:id="echoid-s1539" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1539" xml:space="preserve"> +A continuation from Add MS 6782, f. 445 and f. 446, of Harriot's work on the 'Dati sexti' +from Chapter XIX of Viète's +<emph style="it">Variorum responsorum liber VIII</emph> (1593). +Here he examines Proposition II, part 1. +</s> +<lb/> +<quote xml:lang="lat"> +II. <lb/> +Data differentia duarum perpheriarum, quarum transsinuosæ datam habeant rationem, dantur singulæ. <lb/> +1 Enimvero si differentia fit major circuli quadrante, <lb/> +Erit, <lb/> +Vt transsinuosa differentium primæ ad transsinuosum secundæ, ita transsinuosa complementi differentiæ +ad prosinum complementi differentiaæ plus prosinu secundæ. +</quote> +<lb/> +<quote> +II.Given the difference of two arcs, whose secants are in a given ratio, each is given individually. <lb/> +<lb/>[...]<lb/> <lb/> +1. If the difference is greater than a quadrant, then +as the secant of the first is to the secant of the second, so is the secant of the complement of the difference +to the sine of the complement of the difference plus the sine of the second. +</quote> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head200" xml:space="preserve" xml:lang="lat"> +Vieta. resp. lib. 8. <lb/> +pag. 38. <lb/> +Data differentia duarum perpheriarum +et ratione <emph style="super"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Ψ</mo></mstyle></math></emph> +<lb/>[<emph style="it">tr: +Viète, Responsorum liber VIII, page 38. <lb/> +Given the difference of two arcs, and the ratio of their secants. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1541" xml:space="preserve"> +1. <lb/> +differentia <lb/> +maior quadrante +<lb/>[<emph style="it">tr: +1. difference greater than a quadrant. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1542" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> una peripheria <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> altera <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>i</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>i</mi></mstyle></math> differentia <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>k</mi></mstyle></math> compl. differentiæ <lb/> +<lb/>[...]<lb/> <lb/> +Triangula similia +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> one arc <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> the other <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>i</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>i</mi></mstyle></math> the difference <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>k</mi></mstyle></math> complement of the difference <lb/> +<lb/>[...]<lb/> <lb/> +Similar triangles +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1543" xml:space="preserve"> +primus terminus minor <lb/> +primus term maior +<lb/>[<emph style="it">tr: +the first term less <lb/> +the first term greater +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f447v" o="447v" n="895"/> +<pb file="add_6782_f448" o="448" n="896"/> +<div xml:id="echoid-div289" type="page_commentary" level="2" n="289"> +<p> +<s xml:id="echoid-s1544" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1544" xml:space="preserve"> +A continuation of Harriot's work on the 'Dati sexti' from Chapter XIX of Viète's +<emph style="it">Variorum responsorum liber VIII</emph> (1593). +Here he examines Proposition II, part 2. +</s> +<lb/> +<quote xml:lang="lat"> +II. <lb/> +Data differentia duarum perpheriarum, quarum transsinuosæ datam habeant rationem, dantur singulæ. <lb/> +<lb/>[...]<lb/> <lb/> +2 Et si differentia fit minor quadrante, differentes autem peripheriæ diversæ sint speciei. <lb/> +Erit, <lb/> +Vt transsinuosa primæ ad transsinuosum secundæ, ita transsinuosa complementi differentiæ +ad prosinum secundæ minus prosinu complementi differentiæ. +</quote> +<lb/> +<quote> +II.Given the difference of two arcs, whose secants are in a given ratio, each is given individually. <lb/> +<lb/>[...]<lb/> <lb/> +2. And if the difference is less than a quadrant, and also the signs of each arc are different, then +as the secant of the first is to the secant of the second, so is the secant of the complement of the difference +to the sine of the second minus the sine of the complement of the difference. +</quote> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head201" xml:space="preserve" xml:lang="lat"> +Vieta. resp. lib. 8. <lb/> +pag. 38. <lb/> +Data differentia duarum perpheriarum +et ratione <emph style="super"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Ψ</mo></mstyle></math></emph> +<lb/>[<emph style="it">tr: +Viète, Responsorum liber VIII, page 38. <lb/> +Given the difference of two arcs, and the ratio of their secants. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1546" xml:space="preserve"> +2. <lb/> +Differentia minor <lb/> +quadrante. <lb/> +peripheria <lb/> +una minor, <lb/> +altera maior quadrante +<lb/>[<emph style="it">tr: +2. Difference less than a quadrant; one arc greater than a quadrant, the other less. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1547" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> una peripheria minor quadrante <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> altera peripheria maior quadrante <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>i</mi><mo>=</mo><mi>a</mi><mi>b</mi><mo>=</mo><mi>k</mi><mi>m</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>i</mi></mstyle></math> differentia, minor quadrante <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>n</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>f</mi></mstyle></math> parallelæ <lb/> +Triangula similia +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> one arc, less than a quadrant <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> the other arc, greater than a quadrant <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>i</mi><mo>=</mo><mi>a</mi><mi>b</mi><mo>=</mo><mi>k</mi><mi>m</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>i</mi></mstyle></math> the difference, less than a quadrant <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>n</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>f</mi></mstyle></math> parallels <lb/> +Similar triangles +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f448v" o="448v" n="897"/> +<pb file="add_6782_f449" o="449" n="898"/> +<div xml:id="echoid-div290" type="page_commentary" level="2" n="290"> +<p> +<s xml:id="echoid-s1548" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1548" xml:space="preserve"> +A continuation of Harriot's work on the 'Dati sexti' from Chapter XIX of Viète's +<emph style="it">Variorum responsorum liber VIII</emph> (1593). +Here he examines Proposition II, part 3. +</s> +<lb/> +<quote xml:lang="lat"> +II. <lb/> +Data differentia duarum perpheriarum, quarum transsinuosæ datam habeant rationem, dantur singulæ. <lb/> +<lb/>[...]<lb/> <lb/> +3 Et si denique differentia fit minor quadrante, utraque vero differentium vel minor quadrante, +vel utraque major. Prima autem intelligatur ea ad quam pertinet transsiuosa major. <lb/> +Erit, <lb/> +Vt transsinuosa primæ ad transsinuosum secundæ, ita transsinuosa complementi differentiæ +ad prosinum secundæ minus prosinu complementi differentiæ. <lb/> +Et, <lb/> +Vt transsinuosa primæ ad transsinuosum secundæ, ita transsinuosa complementi differentiæ +ad prosinum primæ minus prosinu complementi differentiæ. +</quote> +<lb/> +<quote> +II.Given the difference of two arcs, whose secants are in a given ratio, each is given individually. <lb/> +<lb/>[...]<lb/> <lb/> +3. If the difference is less than a quadrant, and both arcs are less than a quadrant, or both greater, then +as the secant of the first is to the secant of the second, so is the secant of the complement of the difference +to the sine of the second minus the sine of the complement of the difference; and +as the secant of the first is to the secant of the second, so is the secant of the complement of the difference +to the sine of the first minus the sine of the complement of the difference. +</quote> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head202" xml:space="preserve" xml:lang="lat"> +Vieta. pag. 38.b. resp. lib. 8. <lb/> +Data differentia duarum perpheriarum +et ratione <emph style="super"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Ψ</mo></mstyle></math></emph> +<lb/>[<emph style="it">tr: +Viète, page 38v, Responsorum liber VIII. <lb/> +Given the difference of two arcs, and the ratio of their secants. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1550" xml:space="preserve"> +3. bis +<lb/>[<emph style="it">tr: +3. twofold. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1551" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> una peripheria <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> altera <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>i</mi><mo>=</mo><mi>a</mi><mi>b</mi><mo>=</mo><mi>k</mi><mi>m</mi></mstyle></math> <lb/> +<lb/>[...]<lb/> <lb/> +Ergo. anguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>o</mi><mi>g</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>d</mi></mstyle></math> æquales <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>n</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math> parallelæ <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi><mo>=</mo><mi>k</mi><mi>n</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>c</mi></mstyle></math>, differentia inter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> <lb/> +Triangula similia +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> one arc <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> the other <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>i</mi><mo>=</mo><mi>a</mi><mi>b</mi><mo>=</mo><mi>k</mi><mi>m</mi></mstyle></math> <lb/> +<lb/>[...]<lb/> <lb/> +Therefore angles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>o</mi><mi>g</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>e</mi><mi>d</mi></mstyle></math> are equal <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>n</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math> are parallels <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi><mo>=</mo><mi>k</mi><mi>n</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>c</mi></mstyle></math>, the difference between <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> <lb/> +Similar triangles +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1552" xml:space="preserve"> +Differentia minor <lb/> +quadrante. <lb/> +Utraque peripheria <lb/> +etiam minor, +<lb/>[<emph style="it">tr: +Difference less than a quadrant; each arc also less. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f449v" o="449v" n="899"/> +<pb file="add_6782_f450" o="450" n="900"/> +<div xml:id="echoid-div291" type="page_commentary" level="2" n="291"> +<p> +<s xml:id="echoid-s1553" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1553" xml:space="preserve"> +The reference on this page is to Viète's +<emph style="it">Variorum responsorum liber VIII</emph>, +Chapter 14, Proposition 1. +</s> +<lb/> +<quote xml:lang="lat"> +Propositio I. <lb/> +Datis duabus inæqualibus lineis, una recta, altera circulari, invenire lineam rectam +minorem majore datarum, & majorem minore. +</quote> +<lb/> +<quote> +Given two unequal lines, one straight, the other circular, to find a straight line +less than the greater of those given, and greater than the lesser. +</quote> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head203" xml:space="preserve" xml:lang="lat"> +Resp. lib. 8. cap. 14. prop. 1. pag. 23 +<lb/>[<emph style="it">tr: +Responsorum liber VIII, Chapter 14, Proposition 1, page 23. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1555" xml:space="preserve"> +Datis duabus inæqualibus lineis: una recta, altera circulari: <lb/> +invenire lineam rectam, minorem majore datarum et majorem minore. +<lb/>[<emph style="it">tr: +Given two unequal lines, one straight, the other circular, to find a straight line +less than the greater of those given, and greater than the lesser. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1556" xml:space="preserve"> +Casus primus. +<lb/>[<emph style="it">tr: +First case. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1557" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> recta. maior. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi></mstyle></math> circularis <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math> Excessus <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math> Circulari +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> be the straight line, greater. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi></mstyle></math> the circular line <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math> the excess <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math> the circular line. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1558" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> maior circulari <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> quæcunque maior <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> is greater than the circular line. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> is any quantity greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1559" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>.</mo><mi>G</mi><mo>.</mo><mi>H</mi><mo>-</mo><mi>Z</mi><mo>.</mo><mi>A</mi></mstyle></math> quæsita linea <lb/> +Nam subducendo ab <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>-</mo><mi>Z</mi></mstyle></math>. remanabit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. <lb/> +et a <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>. remanabit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>A</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> igitur est minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>:</mo><mi>G</mi><mo>=</mo><mi>H</mi><mo>-</mo><mi>Z</mi><mo>:</mo><mi>A</mi></mstyle></math>, the line sought. <lb/> +For subtracting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>-</mo><mi>Z</mi></mstyle></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math>, there remains <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>, there remains <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>A</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> is therefore less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1560" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>.</mo><mi>G</mi><mo>.</mo><mi>Z</mi><mo>.</mo><mi>F</mi></mstyle></math> minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. <lb/> +et subducendo: <lb/> +Ergo collata ista proportione cum prima: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>F</mi><mo>=</mo><mi>A</mi></mstyle></math>. <lb/> +Cum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> sit minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>, ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>F</mi></mstyle></math> est Maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math>. <lb/> +Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> est Maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math>. et minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>:</mo><mi>G</mi><mo>=</mo><mi>Z</mi><mo>:</mo><mi>F</mi></mstyle></math> less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. <lb/> +and subtracting: <lb/> +Therefore, combining this proportion with the first, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>F</mi><mo>=</mo><mi>A</mi></mstyle></math>. <lb/> +Since <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>, therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>F</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math>. <lb/> +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math> and less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f450v" o="450v" n="901"/> +<pb file="add_6782_f451" o="451" n="902"/> +<div xml:id="echoid-div292" type="page_commentary" level="2" n="292"> +<p> +<s xml:id="echoid-s1561" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1561" xml:space="preserve"> +This page is a continuation of Add MS 6782, f. 450, on Viète's +<emph style="it">Variorum responsorum liber VIII</emph>, +Chapter 14, Proposition 1. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head204" xml:space="preserve" xml:lang="lat"> +Casus primus. +<lb/>[<emph style="it">tr: +First case. +</emph>]<lb/> +</head> +<pb file="add_6782_f451v" o="451v" n="903"/> +<pb file="add_6782_f452" o="452" n="904"/> +<div xml:id="echoid-div293" type="page_commentary" level="2" n="293"> +<p> +<s xml:id="echoid-s1563" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1563" xml:space="preserve"> +This page is a continuation of Add MS 6782, f. 450 and f. 451, on Viète's +<emph style="it">Variorum responsorum liber VIII</emph>, +Chapter 14, Proposition 1. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head205" xml:space="preserve" xml:lang="lat"> +Resp. lib. 8. cap. 14. prop. 1. pag. 23 +<lb/>[<emph style="it">tr: +Responsorum liber VIII, Chapter 14, Proposition 1, page 23. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1565" xml:space="preserve"> +Casus secundus. +<lb/>[<emph style="it">tr: +Second case. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1566" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> recta. minor. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi></mstyle></math> circularis <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math> Excessus +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> be the straight line, lesser. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi></mstyle></math> the circular line <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math> the excess +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1567" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> minor circulari +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> is less than the circular line. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1568" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>.</mo><mi>G</mi><mo>.</mo><mi>H</mi><mo>+</mo><mi>Z</mi><mo>.</mo><mi>E</mi></mstyle></math> quæsita linea <lb/> +Et subducendo ab <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>+</mo><mi>Z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math>: remanabit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. <lb/> +et ab <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. remanabit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mo>-</mo><mi>G</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> igitur maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>:</mo><mi>G</mi><mo>=</mo><mi>H</mi><mo>+</mo><mi>Z</mi><mo>:</mo><mi>E</mi></mstyle></math>, the line sought. <lb/> +For subtracting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>+</mo><mi>Z</mi></mstyle></math>, there remains <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math>, there remains <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mo>-</mo><mi>G</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> is therefore greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1569" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>.</mo><mi>G</mi><mo>.</mo><mi>Z</mi><mo>.</mo><mi>F</mi></mstyle></math> minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. <lb/> +Et per additione: <lb/> +Ergo, collata ista proportione cum prima: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>+</mo><mi>F</mi><mo>=</mo><mi>E</mi></mstyle></math>. <lb/> +Cum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> sit minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>, ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>+</mo><mi>F</mi></mstyle></math> est minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>+</mo><mi>Z</mi></mstyle></math>. <lb/> +Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> est minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>+</mo><mi>Z</mi></mstyle></math>. et Maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>:</mo><mi>G</mi><mo>=</mo><mi>Z</mi><mo>:</mo><mi>F</mi></mstyle></math> less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. <lb/> +And by addition: <lb/> +Therefore, combining this proportion with the first, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>+</mo><mi>F</mi><mo>=</mo><mi>E</mi></mstyle></math>. <lb/> +Since <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>, therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>+</mo><mi>F</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>+</mo><mi>Z</mi></mstyle></math>. <lb/> +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math> and greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f452v" o="452v" n="905"/> +<pb file="add_6782_f453" o="453" n="906"/> +<div xml:id="echoid-div294" type="page_commentary" level="2" n="294"> +<p> +<s xml:id="echoid-s1570" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1570" xml:space="preserve"> +This page appears to be a continuation of Harriot's work on Add MS 6782, f. 450 and f. 452, +on Proposition 1 from Viète's <emph style="it">Variorum responsorum liber VIII</emph>, Chapter 14. +Here he also refers to Euclid's <emph style="it">Elements</emph>, +Book III, Proposition 16. +</s> +<lb/> +<quote> +III.16 The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle, +and into the space between the straight line and the circumference another straight line cannot be interposed; +further the angle of the semicircle is greater, and the remaining angle less, than any acute rectilineal angle. +</quote> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s1572" xml:space="preserve"> +Si angulus semicirculi sit minor recto rectilineo: dabitur <lb/> +angulus rectilineus maior angulo semicirculi et minor recto <lb/> +rectilineo. Contra Eculidem lib. 3. prop. 16. +<lb/>[<emph style="it">tr: +If the angle in a semicircle is less than a right angle, +there may be found an angle greater than the angle in the semicircle and less than a right angle. +Against Euclid III.16. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1573" xml:space="preserve"> +Si angulus rectus et maior angulo semicirculi. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. <lb/> +Differentia inter angulum rectum et angulum semicurculi. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. <lb/> +Ergo angulus semicirculi. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math>. <lb/> +Et sit aliquis angulus rectilineus maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>H</mi></mstyle></math> +<lb/>[<emph style="it">tr: +Suppose a right angle is greater than the angle in a semicircle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. <lb/> +The difference between the right angle and the angle in the semicircle is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. <lb/> +Therefore the angle in the semicircle is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math>. <lb/> +And let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> be any angle greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1574" xml:space="preserve"> +Tum ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> ita <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>-</mo><mi>Z</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>. +aliquem <emph style="super">rectilineum</emph> angulum ... qui <lb/> +minor est quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> et maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Then as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> is to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>-</mo><mi>Z</mi></mstyle></math> is to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>, another angle which is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> and greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1575" xml:space="preserve"> +Nam subducendo ab <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>-</mo><mi>Z</mi></mstyle></math>: remanebit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math> <lb/> +et ab <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>: remanebit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>A</mi></mstyle></math>. <lb/> +Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>.</mo><mi>G</mi><mo>.</mo><mi>Z</mi><mo>.</mo><mi>G</mi><mo>-</mo><mi>A</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> igitur minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. <lb/> +Et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>.</mo><mi>G</mi><mo>.</mo><mi>Z</mi><mo>.</mo><mi>F</mi></mstyle></math> minorem quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. <lb/> +Ergo subducendo erit <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>.</mo><mi>G</mi><mo>.</mo><mi>H</mi><mo>-</mo><mi>Z</mi><mo>.</mo><mi>G</mi><mo>-</mo><mi>F</mi></mstyle></math> <lb/> +Ergo collata ista proportione cum prima: erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>F</mi><mo>=</mo><mi>A</mi></mstyle></math>. <lb/> +Et: cum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> sit minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>, Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>F</mi></mstyle></math> est Maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math>. <lb/> +Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> est Maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math> et minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Now subtracting <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>-</mo><mi>Z</mi></mstyle></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> there remains <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>; and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>, there remains <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>A</mi></mstyle></math>. <lb/> +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>:</mo><mi>G</mi><mo>=</mo><mi>Z</mi><mo>:</mo><mi>G</mi><mo>-</mo><mi>A</mi></mstyle></math>, therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. <lb/> +And <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>:</mo><mi>G</mi><mo>=</mo><mi>Z</mi><mo>:</mo><mi>F</mi></mstyle></math> less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. <lb/> +Therefore, subtracting, <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mo>:</mo><mi>G</mi><mo>=</mo><mi>H</mi><mo>-</mo><mi>Z</mi><mo>:</mo><mi>G</mi><mo>-</mo><mi>F</mi></mstyle></math> <lb/> +Therefore, combining this proportion with the first, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>F</mi><mo>=</mo><mi>A</mi></mstyle></math>. <lb/> +And since <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>, therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>F</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math>. <lb/> +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mo>-</mo><mi>Z</mi></mstyle></math> and less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f453v" o="453v" n="907"/> +<pb file="add_6782_f454" o="454" n="908"/> +<div xml:id="echoid-div295" type="page_commentary" level="2" n="295"> +<p> +<s xml:id="echoid-s1576" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1576" xml:space="preserve"> +The reference on this page is to Viète's +<emph style="it">Variorum responsorum liber VIII</emph>, +Chapter 14, Proposition 3. +</s> +<lb/> +<quote xml:lang="lat"> +Propositio III. <lb/> +Circulo dato, & linea recta in eo inscripta, quæ diametro minor existat, +& alia insuper quæ circulum tangat in inscriptæ termino, educere lineam e centro +ita secantem circulum & ipsam tangentem, ut pars educate e centro interjacens +inter cirumferentiam & inscriptam se habeat ad partem tangentis quæ est inter contactum & ipsam eductam, +sicut dimidia inscripta ad majorem ea quæ ex centro inscriptam illam bisariam secat. +</quote> +<lb/> +<quote> +Given a circle and a straight line inscribed in it, less than the diameter, +and also a line that touches the circle at the end of the inscribed line, +to draw a line from the centre cutting the circle and the tangent, so that the part drawn from the centre +lying between the circumference and the inscribed line is to the part of the tangent +between the contact and the drawn line as half the inscribed line to a line longer than that +drawn from the centre bisecting the inscribed line. +</quote> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head206" xml:space="preserve" xml:lang="lat"> +In Caput 14. Responsorum vieta prop. 3. pa. 24 +<lb/>[<emph style="it">tr: +From Chapter 14 of Viète's Responsorum, Proposition 3, page 24. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1578" xml:space="preserve"> +et subducendo: +<lb/>[<emph style="it">tr: +and subtracting: +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f454v" o="454v" n="909"/> +<pb file="add_6782_f455" o="455" n="910"/> +<div xml:id="echoid-div296" type="page_commentary" level="2" n="296"> +<p> +<s xml:id="echoid-s1579" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1579" xml:space="preserve"> +The reference on this page is to Viète's +<emph style="it">Variorum responsorum liber VIII</emph>, Chapter 9, Proposition 13. +</s> +<lb/> +<quote xml:lang="lat"> +Et si fuerint lineæ quotcunque sese excedentes, fit autem prima excessui aequalis, +fiunt ab iis quatuor solida coninue proportionalia, qualia sequntur. <lb/> +Primum, Cubus minimæ. <lb/> +Secundum, Cubus compositæ ex maxima & minima, multaus adgregato cuborum minimæ & maximæ. <lb/> +Tertium, Adgregatum cuborume singulis ter duodecuplum. <lb/> +Quartum, Cubus compositæ ex omnibus sextuplae. +</quote> +<lb/> +<quote> +And if there are any number of lines exceeding each other, and moreover the first differences are equal, +there arise from them four solids in continued proportion, which are as follows. <lb/> +First, the cube of the least. <lb/> +Second, the cube of the sum of the greatest and least, reduced by the sum of the cubes of the least and the greatest. <lb/> +Third, the sum of the cubes of each taken 36 times. <lb/> +Fourth, the cube of the sum of all, taken six times. +</quote> +<lb/> +<s xml:id="echoid-s1580" xml:space="preserve"> +Note that in his initial calculations, expressed in terms of geometric solids, +Harriot retains homogeneity by writing, for example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>1</mn></mstyle></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>×</mo><mn>1</mn></mstyle></math>. +In the second column, where he moves from a geometric to arithmetic interpretation, +he simplifies the notation by dropping the 1s. +</s> +<lb/> +<s xml:id="echoid-s1581" xml:space="preserve"> +For Harriot's teaching on progressions, see in particular Add MS 6782, f. 107 to f. 146v. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head207" xml:space="preserve" xml:lang="lat"> +In propositione 13, cap.9. lib. 8. respons. pag. 14. b. Vietæ +<lb/>[<emph style="it">tr: +From Proposition 13, Chapter 9, Liber VIII responsorum, page 14v, of Viète +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1583" xml:space="preserve"> +propositio <lb/> +Si fuerint lineæ quotcunque sese excedentes, fit autem prima excessui aequalis, <lb/> +fiunt ab iis quatuor solida coninue proportionalia, qualia sequntur. <lb/> +Primum, Cubus minimæ. <lb/> +Secundum, Cubus compositæ ex maxima et minima, multaus aggregato <lb/> +cuborum minimæ et maximæ. <lb/> +Tertium, Adgregatum cuborume singulis ter duodecuplum. <lb/> +Quartum, Cubus compositæ ex omnibus sextuplae. +<lb/>[<emph style="it">tr: +If there are any number of lines exceeding each other, and moreover the first differences are equal, +there arise from them four solids in continued proportion, which are as follows. <lb/> +First, the cube of the least. <lb/> +Second, the cube of the sum of the greatest and least, reduced by the sum of the cubes of the least and the greatest. <lb/> +Third, the sum of the cubes of each taken 36 times. <lb/> +Fourth, the cube of the sum of all, taken six times. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1584" xml:space="preserve"> +Sit minima 1. maxima 4. Summa omnia sextupla sit 60. et sunt continuo <lb/> +proportionalia solida. 1. 60. 360. 21600. +<lb/>[<emph style="it">tr: +Let the least quantity be 1, the greatest 4. Six times the sum of all of them is 60, +and the proportional solids are 1, 60, 360, 21600. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1585" xml:space="preserve"> +Sit prima linea et excessus 1. <lb/> +et numerus linearum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> <lb/> +Summa <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>, sive triangulus numerus <lb/> +ipsius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> erit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi><mi>n</mi><mo>+</mo><mn>1</mn><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>. <lb/> +cuius quadratum erit: <lb/> +atque hoc est per doctrina progressionum <lb/> +summa cuborum singulis <lb/> +Eis ter duodecuplum erit: <lb/> +Hoc est: <lb/> +pro 3<emph style="super">o</emph> solido +<lb/>[<emph style="it">tr: +Let the first line and the excess be 1, and <lb/> the number of lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>. <lb/> +Their sum, or triangular number, will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi><mi>n</mi><mo>+</mo><mn>1</mn><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>, whose square is: <lb/> +and by the doctrine of progressions, this is the sum of individual cubes. <lb/> +Three times their 12-tuple will be: <lb/> +That is: <lb/> +for the 3rd solid. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1586" xml:space="preserve"> +Composita ex maxima et minima <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>+</mo><mn>1</mn></mstyle></math> <lb/> +Eius cubus erit <lb/> +Cubis multatis; erit <lb/> +pro 2<emph style="super">o</emph> solido +<lb/>[<emph style="it">tr: +The sum of the greatest and least is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>+</mo><mn>1</mn></mstyle></math>. <lb/> +Its cube will be: </emph>]<lb/> +The cubes having been subtracted this will be: <lb/> +for the 2nd solid. +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1587" xml:space="preserve"> +Composita ex omnibus ut supra <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi><mi>n</mi><mo>+</mo><mn>1</mn><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> <lb/> +Eius sextupla: <lb/> +Hoc est: <lb/> +Eius quadratum: <lb/> +Eius cubus: <lb/> +pro 4<emph style="super">o</emph> <lb/> +solido <lb/> +Quadratum si <emph style="st">dividatur</emph> <emph style="super">multiplicatur</emph> per 1. +<emph style="st">[???]</emph> <emph style="super">primam lineam</emph> <lb/> +faciet <emph style="st">secundum</emph> <emph style="super">tertium</emph> solidum ut supra. +<lb/>[<emph style="it">tr: +The sum of all, as above, is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi><mi>n</mi><mo>+</mo><mn>1</mn><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>. <lb/> +Six time this: <lb/> +That is: <lb/> +Its square: <lb/> +Its cube: <lb/> +for the 4th solid. <lb/> +The square, if the first line is multiplied by 1, makes the third solid, as above. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1588" xml:space="preserve"> +Solida continue proportionales +<lb/>[<emph style="it">tr: +The continually proportional solids +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1589" xml:space="preserve"> +proportionalis operatio, dato primo <lb/> +et secundo, inveniet 3<emph style="super">um</emph> et 4<emph style="super">tum</emph> <lb/> +sub maiori formam notari sed <lb/> +per reductionem; æqualia sunt illis. +<lb/>[<emph style="it">tr: +The operation of proportion, given the first and second, will find the 3rd and 4th, under a greater form of notation, +but by reduction, they are equal. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1590" xml:space="preserve"> +pro numeris vel numeris <lb/> +solidarum talis potest <lb/> +esse notatio: et si fit <lb/> +in forma heterogenea. Vel: <lb/> +ennuntiari potest per lineas. +<lb/>[<emph style="it">tr: +for numbers, or solid numbers, the notation my be thus; and if done in heteregenous form. +Or it may be expressed in lines. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1591" xml:space="preserve"> +datis primo et secundo +proportionalis operatio 3<emph style="super">um</emph> & 4<emph style="super">um</emph> <lb/> +invenient ut sunt. +<lb/>[<emph style="it">tr: +given the first and second, by the operation of proportions, the 3rd and 4th may be found and are: +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1592" xml:space="preserve"> +Nota, quod <lb/> +etiam proportio potest <lb/> +inveniri per plana <lb/> +per lineas <lb/> +per plano-plana &c. <lb/> +et eadem modi demonstrari. Etiam: <lb/> +proportionalia possunt fieri plano numero <lb/> +ad libitum secundum nostram doctrinam de <lb/> +progressionibus, quæ per traditiones veterum <lb/> +fieri non potuit. +<lb/>[<emph style="it">tr: +Note, that the proportion may also be found by planes, lines, plano-planes, etc. +and demosntrated by the same method. Also, the proportionals may arise from plane numbers at will +according to my doctrine of progressions, which by old teachings could not be done. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f455v" o="455v" n="911"/> +<pb file="add_6782_f456" o="456" n="912"/> +<div xml:id="echoid-div297" type="page_commentary" level="2" n="297"> +<p> +<s xml:id="echoid-s1593" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1593" xml:space="preserve"> +This page contains Harriot's working of Zetetic 10, the last from Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book I. +</s> +<lb/> +<quote xml:lang="lat"> +Zeteticum X <lb/> +Invenire duo latera, quorum differentia fit ea quæ præscribitur, +& præterea præfinitae unciæ primi, multatæ præfinitis unciis secundi, +æquent differentiam quoque inter eas datam. +</quote> +<lb/> +<quote> +To find two lines, whose difference is prescribed, +and also such that a fixed part of the first taken from a fixed part of the second +is likewise equal to a given difference. +</quote> +<lb/> +<s xml:id="echoid-s1594" xml:space="preserve"> +Viète used the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the difference between the two lines, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the first part to the first line, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the second part to the second line, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> for the gvien difference, +and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> for the parts of the first and second lines. +Harriot worked through the three cases given by Viète, using his own notation. <lb/> +The work is continued using numbers on Add MS 6782, f. 459. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head208" xml:space="preserve" xml:lang="lat"> +Zetet. lib. 1. Zet. 10. et primi lib. ultimum Sept. 6. +<lb/>[<emph style="it">tr: +Zetetica, Book I, Zetetic 10, and the last of the first book. September 6 +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1596" xml:space="preserve"> +Invenire duo latera, quorum differentia sit ea quæ præscribitur, et præterea <lb/> +præfinitæ unciæ lateris primi multatæ præfinitis unciis secundi, æquent <lb/> +differentiam quoque inter eas datam. +<lb/>[<emph style="it">tr: +To find two lines, whose difference is prescribed, +and also such that a fixed part of the first subtracted from a fixed part of the second +is likewise equal to a given difference. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1597" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. data differentia 2<emph style="super">orum</emph> laterum. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. primum latus. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. secundum latus. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. portio primi lateri. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math> <lb/> +hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>a</mi><mo>,</mo><mi>o</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. portio secundi lateris <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>e</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math> <lb/> +hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>e</mi><mo>,</mo><mi>u</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. differentia data <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>a</mi><mo>-</mo><mi>e</mi></mstyle></math> <lb/> +unde: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>+</mo><mi>e</mi><mo>=</mo><mi>a</mi></mstyle></math>. et: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>h</mi><mo>=</mo><mi>e</mi></mstyle></math>. <lb/> +Quæruntur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, the given difference between the two lines. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, the first line <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, the second line. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, the portion of the first line. <lb/> +The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math>, <lb/> +that is: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>a</mi><mo>:</mo><mi>o</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, the portion of the second line <lb/> +The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>e</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math>, <lb/> +that is: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>e</mi><mo>:</mo><mi>u</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>, the given difference, equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>e</mi></mstyle></math>. <lb/> +whence: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>+</mo><mi>e</mi><mo>=</mo><mi>a</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>h</mi><mo>=</mo><mi>e</mi></mstyle></math>. <lb/> +There are sought <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1598" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. primum latus intelligitur maius duarum <lb/> +vel minus; sive ab eo exigantur unciæ <lb/> +maioris vel minoris. +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, the first line can be understood to be the greater of the two, <lb/> +or the smaller; whether from it are taken greater fractions or smaller. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1599" xml:space="preserve"> +1. casus. sit primum latus maius: <lb/> +et exigantur ab eo maioris unciæ. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> portio primi lateris. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math> portio secundi lateris. +<lb/>[<emph style="it">tr: +Case 1. Let the first line be greater, and from it are taken greater fractions. <lb/> +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be the portion of the first line, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math> the portion of the second line. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1600" xml:space="preserve"> +Porrò, sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, portio <lb/> +secundi lateris. +<lb/>[<emph style="it">tr: +Further, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the portion of the second line. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1601" xml:space="preserve"> +2. casus. sit primum latus maius <lb/> +et exigantur ab eo minoris unciæ. <lb/> +hoc est ponatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Case 2. Let the first line be greater, and from it are taken smaller fractions, +that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is supposed less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1602" xml:space="preserve"> +Porrò, sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, portio <lb/> +secundi lateris. +<lb/>[<emph style="it">tr: +Further, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the portion of the second line. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1603" xml:space="preserve"> +Nota. <lb/> +Hinc apparet quod <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> debet esse. <lb/> +In primo casu maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, ac etiam quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> <lb/> +In secundo casu minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. <lb/> +In tertio maior vel minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Note. <lb/> +Here it is clear what <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> must be. <lb/> +In the first case, greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> and also than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. <lb/> +In the second case, less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. <lb/> +In the third, greater or less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1604" xml:space="preserve"> +3. casus. sit primum latus minus <lb/> +et exigantur ab eo maioris unciæ. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> portio primi lateris. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math> portio secundi lateris. +<lb/>[<emph style="it">tr: +Case 3. Let the first line be smaller, and from it ar taken greater fractions. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> is the portion of the first line, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math> the portion of the second line. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1605" xml:space="preserve"> +Porrò, sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, portio secundi lateris. +<lb/>[<emph style="it">tr: +Further, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the portion of the second line. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1606" xml:space="preserve"> +* Nota. <lb/> +Hinc facile apparet quod minores <lb/> +unciæ non possunt exigi à primo latere <lb/> +cum sit minus. Atque ideo non <lb/> +datur quartus casus. +<lb/>[<emph style="it">tr: +* Note. <lb/> +Here is is easily seen that smaller fractions cannot be taken from the first line since it is smaller. +And therefore therefore no fourth case is given. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f456v" o="456v" n="913"/> +<pb file="add_6782_f457" o="457" n="914"/> +<div xml:id="echoid-div298" type="page_commentary" level="2" n="298"> +<p> +<s xml:id="echoid-s1607" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1607" xml:space="preserve"> +This page contains the continuation of Harriot's working of Zetetic 7 from Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book I. +</s> +<lb/> +<quote xml:lang="lat"> +Zeteticum VII <lb/> +Datum latus ita secare, ut praefinitae unciae unius segmenti, adjunctae praefinitis unciis alterius: +aequent summam praescriptam. +</quote> +<lb/> +<quote> +To cut a given line in such a way that a fixed part of one segment, added to a fixed part of the other, +is equal to a prescribed sum. +</quote> +<lb/> +<s xml:id="echoid-s1608" xml:space="preserve"> +Viète used the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the whole line, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the first part to the first segment, +and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the second part to the second segment, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> for the prescribed sum, +and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> for the unknown parts of the first and second segments. +Harriot followed Viète's method on Add MS 6782, f. 458. +Here he works the same problem using numbers instead of lengths. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head209" xml:space="preserve" xml:lang="lat"> +Zet. lib. 1. Zet. 7. +<lb/>[<emph style="it">tr: +Zetetica, Book I, Zetetic 7 +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1610" xml:space="preserve"> +Dividere numerum <emph style="super">datum</emph> in duas partes, ita ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> +<emph style="super">unius</emph> primæ partis additæ, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math>, secundæ: æquet <lb/> +summam præscriptam. oportuit ut summa præscripta sit +minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> numeri dati et maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math>. +<lb/>[<emph style="it">tr: +To divide a given number into two parts, so that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of one, the first part, +added to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math> of the second equals a prescribed sum; +the prescribed sum must be less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of the given number and greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1611" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. sit numerus datus. 60 <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. igitur 15. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> erit 10. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> summa præscribenda debet <lb/> +esse maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, et minor <lb/> +quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. sit ergo 12. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the given number, 60. Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is 15 and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> will be 10. <lb/> +The prescribed sum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> must be grater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> and less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, therefore let it be 12. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1612" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. prima pars. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> primæ partis. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> be the first part, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> a <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of the first part. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1613" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. secunda pars. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>: eius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math> +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math> be the second part, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> a <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math> of it. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1614" xml:space="preserve"> +hac analogia ita solvitur problema. <lb/> +<lb/>[...]<lb/> +Ergo. 36. erit secunda pars. <lb/> +6. erit eius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math>. <lb/> +6. est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mi>p</mi><mi>r</mi><mi>i</mi><mi>m</mi><mo>æ</mo></mstyle></math> <lb/> +quæ æquat 12. +<lb/>[<emph style="it">tr: +The problem is thus solved by this ratio. <lb/> +<lb/>[...]<lb/> <lb/> +Therefore 36 will be the second part. <lb/> +6 is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math> of it, <lb/> +6 is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of the first part <lb/> +which makes 12. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1615" xml:space="preserve"> +aliter. <lb/> +<lb/>[...]<lb/> +Ergo. 24. erit prima pars. <lb/> +6. eius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>. <lb/> +6. est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac><mi>s</mi><mi>e</mi><mi>c</mi><mi>u</mi><mi>n</mi><mi>d</mi><mo>æ</mo></mstyle></math> +<lb/>[<emph style="it">tr: +Another way <lb/> +<lb/>[...]<lb/> <lb/> +Therefore 24 will be the first part. <lb/> +6 is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of it, <lb/> +6 is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math> of the second part +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1616" xml:space="preserve"> +Aliud exemplum +<lb/>[<emph style="it">tr: +Another example +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f457v" o="457v" n="915"/> +<pb file="add_6782_f458" o="458" n="916"/> +<div xml:id="echoid-div299" type="page_commentary" level="2" n="299"> +<p> +<s xml:id="echoid-s1617" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1617" xml:space="preserve"> +This is the first of several pages in which Harriot worked on Zetetica 7 to 10 from Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book I. +Harriot at first treated the problems exactly as Viète had done, using the geometric language in which the word +<foreign xml:lang="lat">latus</foreign> represents an unknown line, side, or root. +However, in each case Harriot then switched from geometry to arithmetic, +treating the known and unkown quantities as numbers rather than geometrical quantities. +</s> +<lb/> +<s xml:id="echoid-s1618" xml:space="preserve"> +This page contains Harriot's working of Zetetic 7. +</s> +<lb/> +<quote xml:lang="lat"> +Zeteticum VII <lb/> +Datum latus ita secare, ut præfinitæ unciæ unius segmenti, adjunctae præfinitis unciis alterius: +æquent summam præscriptam. +</quote> +<lb/> +<quote> +To cut a given line in such a way that a fixed part of one segment, added to a fixed part of the other, +is equal to a prescribed sum. +</quote> +<lb/> +<s xml:id="echoid-s1619" xml:space="preserve"> +Viète used the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the whole line, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the first part to the first segment, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the second part to the second segment, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> for the prescribed sum, +and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> for the parts of the first and second segments. +Harriot repeated Viète's working in his own notation, and also added some variants of his own. +The numbers at the bottom of the page are taken from Viète. +On Add MS 6782, f. 457, Harriot worked the same problem with further numerical examples. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head210" xml:space="preserve" xml:lang="lat"> +Lib. 1. Zetet. <lb/> +Zet. 7. +<lb/>[<emph style="it">tr: +Zetetica, Book I, Zetetic 7 +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1621" xml:space="preserve"> +Datum latus ita secare, ut præfinitæ unciæ unius segmenti, adjunctae præfinitis unciis alterius: +æquent summam præscriptam. +<lb/>[<emph style="it">tr: +To cut a given line in such a way that a fixed part of one segment, added to a fixed part of the other, +is equal to a prescribed sum. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1622" xml:space="preserve"> +sit datum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. latus. <lb/> +et duo segmenta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math> <lb/> +portio primæ segmenti <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> <lb/> +ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>: ita debet esse: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> <lb/> +summa præscripta sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> <lb/> +Ergo portio 2<emph style="super">i</emph> segmenti erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math>. <lb/> +ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>: ita debet esse: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. <lb/> +Quæretur iam: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, et, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, segmenta <lb/> +et segmentorum portiones. +<lb/>[<emph style="it">tr: +Let the given line be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, and the two segments <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. <lb/> +The portion of the first segment is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>; as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, so must be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. <lb/> +The prescribed sum is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. <lb/> +Therefore the portion of the second segment will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math>. <lb/> +As <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> is to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, so must be % h - a <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi><mi>o</mi></mstyle></math> u <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>.</mo></mstyle></math><lb/> +There are now sought <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, the segments, and the portions of the segments. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1623" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> minor <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> et maior <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> and greater then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1624" xml:space="preserve"> +Aliter. <lb/> +Sit portio secundæ segmenti, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> <lb/> +Ergo portio primæ segmenti <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>e</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Another way. <lb/> +Let the portion of the second segment be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. <lb/> +Therefore the portion of the first segment is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>e</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1625" xml:space="preserve"> +Dantur etiam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, per superiores analogias. +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math> are also given by the above ratios. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1626" xml:space="preserve"> +Additio nostra. Aliter. +<lb/>[<emph style="it">tr: +An addition of my own. Another way. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1627" xml:space="preserve"> +Aliter. +<lb/>[<emph style="it">tr: +Another way. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1628" xml:space="preserve"> +Nota. <lb/> +Etsi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, sit maior <emph style="super">quam</emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>; est [???] <lb/> +minor <emph style="super">quam</emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> per suppositione: <lb/> +Sed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> est præscribenda ut <lb/> +sit minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, et maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. <lb/> +ut apparet per inventis analogijs. +<lb/>[<emph style="it">tr: +Note. <lb/> +Although <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, it is [???] less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> by supposition. <lb/> +But <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is prescribed so that it is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> and greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, +as is clear from the ratios found. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f458v" o="458v" n="917"/> +<pb file="add_6782_f459" o="459" n="918"/> +<div xml:id="echoid-div300" type="page_commentary" level="2" n="300"> +<p> +<s xml:id="echoid-s1629" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1629" xml:space="preserve"> +This page contains Harriot's working of Zetetic 10, the last from Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book I. +</s> +<lb/> +<quote xml:lang="lat"> +Zeteticum X <lb/> +Invenire duo latera, quorum differentia fit ea quæ præscribitur, +& præterea præfinitae unciæ primi, multatæ præfinitis unciis secundi, +æquent differentiam quoque inter eas datam. +</quote> +<lb/> +<quote> +To find two lines, whose difference is prescribed, +and also such that a fixed part of the first taken from a fixed part of the second +is likewise equal to a given difference. +</quote> +<lb/> +<s xml:id="echoid-s1630" xml:space="preserve"> +Viète used the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the difference between the two lines, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the first part to the first line, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the second part to the second line, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> for the gvien difference, +and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> for the parts of the first and second lines. <lb/> +Harriot followed Viète's method on Add MS 6782, f. 456. +Here he works the same problem using numbers instead of lengths. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head211" xml:space="preserve" xml:lang="lat"> +Zetet. lib. 1. Zet. 10. et ultimum. <lb/> +Exempla in numeris. +<lb/>[<emph style="it">tr: +Zetetica, Book I, Zetetic 10, and the last. <lb/> +Examples in numbers. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1632" xml:space="preserve"> +1. Casus. +<lb/>[<emph style="it">tr: +Case 1. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1633" xml:space="preserve"> +3. Casus. Ubi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> est maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Case 3. Where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1634" xml:space="preserve"> +3. Casus. Ubi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> est minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Case 3. Where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1635" xml:space="preserve"> +2. Casus. +<lb/>[<emph style="it">tr: +Case 2. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1636" xml:space="preserve"> +Additio. 3. Casus. Ubi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> et minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +An addition. Case 3. Where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> and less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1637" xml:space="preserve"> +Si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> sit æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> non variat casum. +<lb/>[<emph style="it">tr: +If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> the case does not change. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f459v" o="459v" n="919"/> +<pb file="add_6782_f460" o="460" n="920"/> +<div xml:id="echoid-div301" type="page_commentary" level="2" n="301"> +<p> +<s xml:id="echoid-s1638" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1638" xml:space="preserve"> +This page contains Harriot's working of Zetetic 9 from Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book I. +</s> +<lb/> +<quote xml:lang="lat"> +Zeteticum IX <lb/> +Invenire duo latera, quorum differentia sit ea quæ præscribitur, +& præterea præfinitæ unciæ lateris unius, adjectæ præfinitis unciis alterius, +æquabunt summam præscriptam. +</quote> +<lb/> +<quote> +To find two lines, whose difference is prescribed, +and also such that a fixed part of one line added to a fixed part of the other is equal to a prescibed sum. +</quote> +<lb/> +<s xml:id="echoid-s1639" xml:space="preserve"> +Viète used the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the difference between the two lines, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the first part to the first line, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the second part to the second line, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> for the prescribed sum, +and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> for the parts of the first and second lines. +Harriot repeated Viète's working in his own notation, and also added some variants of his own. <lb/> +The work is continued using numerical examples on Add MS 6782, f. 461. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head212" xml:space="preserve" xml:lang="lat"> +Zetet. Lib. 1. Zet. 9. +<lb/>[<emph style="it">tr: +Zetetica, Book I, Zetetic 9 +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1641" xml:space="preserve"> +Invenire duo latera, quorum differentia sit ea quæ præscribitur, et præterea <lb/> +præfinitæ unciæ lateris unius adjectæ præfinitis unciis alterius: æquabunt <lb/> +summam præscriptam. +<lb/>[<emph style="it">tr: +To find two lines, whose difference is prescribed, +and also such that a fixed part of one line added to a fixed part of the other is equal to a prescribed sum. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1642" xml:space="preserve"> +Nota. <lb/> +Summa præscripta <lb/> +videlicet <emph style="st">debe</emph><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> debet <lb/> +esse maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, ut <lb/> +per postremis <emph style="st">notat</emph> infra <lb/> +notatis analogijs appa-<lb/> +rebit <emph style="super">scilicet</emph> in primo casu. <lb/> +In secundo casu <lb/> +primum latus ponitur <lb/> +minus, oportet ut <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> fit maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Note. <lb/> +The prescribed sum, namely <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>, must be grater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, +as is apparent from the ratio written afterwards below in the first case. <lb/> +In the second case, where the first line is supposed smaller, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> must be greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1643" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. data differentia 2<emph style="super">orum</emph> laterum. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. primum latus. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. secundum latus. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. portio primi lateri. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math> <lb/> +Hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>a</mi><mo>,</mo><mi>o</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. portio secundi lateris <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>e</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math> <lb/> +Hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>e</mi><mo>,</mo><mi>u</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. Summa præscripta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>a</mi><mo>+</mo><mi>e</mi></mstyle></math> <lb/> +Quæruntur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>: et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, the given difference between the two lines. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, the first line <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, the second line. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, the portion of the first line. <lb/> +The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math> <lb/> +That is: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>a</mi><mo>:</mo><mi>o</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, the portion of the second line <lb/> +The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>e</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math> <lb/> +That is: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>e</mi><mo>:</mo><mi>u</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>, the prescribed sum, equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>+</mo><mi>e</mi></mstyle></math>. <lb/> +There are sought <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1644" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. primum latus intelligitur maius vel minus. <lb/> +primo casu intelligitur maius. +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, the first line can be understood to greater or smaller. <lb/> +In the first case it is understood to be greater. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1645" xml:space="preserve"> +Tum: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> portio primi lateris. <lb/> +Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math> portio secundi lateris. +<lb/>[<emph style="it">tr: +Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> is the portion of the first line. <lb/> +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math> is the portion of the second line. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1646" xml:space="preserve"> +Porrò, sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, portio <lb/> +secundi lateris. +<lb/>[<emph style="it">tr: +Further, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> is the portion of the second line. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1647" xml:space="preserve"> +Secundo casu primum segmentum <lb/> +intelligitur minus. <lb/> +Ergo secundi segmenti erit maius. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> portio secundi lateris. <lb/> +ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>e</mi></mstyle></math> portio primi lateris, et minoris. +<lb/>[<emph style="it">tr: +In the second case the first line is understood to be smaller. <lb/> +Therefore the second line will be greater. <lb/> +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the portion of the second line. <lb/> +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>e</mi></mstyle></math> is the portion of the first line, and smaller. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1648" xml:space="preserve"> +Porrò, sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, portio <lb/> +primi lateris minoris. +<lb/>[<emph style="it">tr: +Further, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be the portion of the first, smaller line. </emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1649" xml:space="preserve"> +Additio nostra pro <lb/> +secundus casus aliter. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, portio primi lateris et minoris <lb/> +Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math> portio secundi lateris, et maioris. <lb/> +Quoniam: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>a</mi><mo>,</mo><mfrac><mrow><mi>b</mi><mi>a</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math> latus minus. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>h</mi><mo>-</mo><mi>a</mi><mo>,</mo><mfrac><mrow><mi>b</mi><mi>h</mi><mo>-</mo><mi>b</mi><mi>a</mi></mrow><mrow><mi>f</mi></mrow></mfrac></mstyle></math> latus maius. +<lb/>[<emph style="it">tr: +An addition of my own for the second case another way. <lb/> +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be the portion of the first and smaller line. <lb/> +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math> is the portion of the second and greater line. <lb/> +Since: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>a</mi><mo>,</mo><mfrac><mrow><mi>b</mi><mi>a</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math> for the smaller line. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>h</mi><mo>-</mo><mi>a</mi><mo>,</mo><mfrac><mrow><mi>b</mi><mi>h</mi><mo>-</mo><mi>b</mi><mi>a</mi></mrow><mrow><mi>f</mi></mrow></mfrac></mstyle></math> for the greater line. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1650" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> portio secundi lateris et maioris +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the portion of the second and greater line. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1651" xml:space="preserve"> +Nota. <lb/> +Alia etiam est <emph style="st">[???]</emph> <emph style="ins ">zetesis supra</emph> ad <lb/> +investigandum secundum portione in <lb/> +primo casu, videlicet ea quæ <lb/> +est primi lateris. +<lb/>[<emph style="it">tr: +Note. <lb/> +The zetesis above is also another for investigating the second portion in the first case, +namely that which is the first line. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f460v" o="460v" n="921"/> +<pb file="add_6782_f461" o="461" n="922"/> +<div xml:id="echoid-div302" type="page_commentary" level="2" n="302"> +<p> +<s xml:id="echoid-s1652" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1652" xml:space="preserve"> +This page continues Harriot's work on Zetetic 9 from Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book I. +</s> +<lb/> +<quote xml:lang="lat"> +Zeteticum IX <lb/> +Invenire duo latera, quorum differentia sit ea quæ præscribitur, +& præterea præfinitæ unciæ lateris unius, adjectæ præfinitis unciis alterius, +æquabunt summam præscriptam. +</quote> +<lb/> +<quote> +To find two lines, whose difference is prescribed, +and also such that a fixed part of one line added to a fixed part of the other is equal to a prescibed sum. +</quote> +<lb/> +<s xml:id="echoid-s1653" xml:space="preserve"> +Viète used the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the difference between the two lines, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the first part to the first line, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the second part to the second line, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> for the prescribed sum, +and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> for the parts of the first and second lines. <lb/> +Harriot followed Viète's method on Add MS 6782, f. 462. +Here he works the same problem using numbers instead of lengths. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head213" xml:space="preserve" xml:lang="lat"> +Zetet. Lib. 1. Zet. 9. +<lb/>[<emph style="it">tr: +Zetetica, Book I, Zetetic 9 +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1655" xml:space="preserve"> +Invenire duas numerus quorum differentia sit 84, et præterea <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> <emph style="st">unius</emph> primi <lb/> +numeri adjecta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> alterius æquabit summam præscriptam. oportet summam præscriptam <lb/> +esse maiorem quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> dictæ differentiæ si primum latus sit maius: sed si minus <lb/> +esse maiorem quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>. +<lb/>[<emph style="it">tr: +To find two numbers whose dfference is 84, and also such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> of the first number +added to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of the other will be equal to a prescribed sum; +the prescribed sum must be greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> of the said difference if the first root is the greater, +but if it is the smaller, it must be greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1656" xml:space="preserve"> +vel: Invenire duas numerus quorum differentia sit 84, et pæterea <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> primi numeri <lb/> +adjecta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> secunda æquabit summam præscriptam. oportet summam præscriptam <lb/> +esse maiorem quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> dictæ differentiæ si primum latus sit maius: sed si minus <lb/> +esse maiorem quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math>. +<lb/>[<emph style="it">tr: +Or: To find two numbers whose dfference is 84, and also such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of the first number +added to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> of the secon will be equal to a prescribed sum; +the prescribed sum must be greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of the said difference if the first root is the greater, +but if it is the smaller, it must be greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1657" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. differntia numerourm. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. primus numerus <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. secundus numerus. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. portio primi numeri. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. portio 2<emph style="super">i</emph> numeri <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>e</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> summa præscripta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>a</mi><mo>+</mo><mi>e</mi></mstyle></math> <lb/> +Quæruntur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, the difference between the numbers. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, the first number <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, the second number. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, the portion of the first number. <lb/> +The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, the portion of the first number. <lb/> +The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>e</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>, the prescribed sum, equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>+</mo><mi>e</mi></mstyle></math>. <lb/> +There are sought <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1658" xml:space="preserve"> +Primum latus maius. Portio maiorum uncium. +<lb/>[<emph style="it">tr: +The first root greater. Portion greater than the fraction. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1659" xml:space="preserve"> +Primum latus minus. Portio maiorum uncium. +<lb/>[<emph style="it">tr: +The first root smaller. Portion greater than the fraction. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1660" xml:space="preserve"> +Primum latus maius. Portio minorum uncium. +<lb/>[<emph style="it">tr: +The first root greater. Portion less than the fraction. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1661" xml:space="preserve"> +Primum latus minus. Portio minorum uncium. +<lb/>[<emph style="it">tr: +The first root smaller. Portion less than the fraction. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1662" xml:space="preserve"> +Primum latus minus. Ubi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> est maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> et minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +The first root smaller, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> but less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f461v" o="461v" n="923"/> +<pb file="add_6782_f462" o="462" n="924"/> +<div xml:id="echoid-div303" type="page_commentary" level="2" n="303"> +<p> +<s xml:id="echoid-s1663" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1663" xml:space="preserve"> +This page continues Harriot's working of Zetetic 8 from Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book I. +</s> +<lb/> +<quote xml:lang="lat"> +Zeteticum VIII <lb/> +Datum latus ita secare, ut præfinitae unciae segmenti, multatæ præfinitis unciis secundi segmenti: +æquent differentiam praescriptam. +</quote> +<lb/> +<quote> +To cut a given line in such a way that a fixed part of one segment, subtracted from a fixed part of the second segment, +is equal to a prescribed difference. +</quote> +<lb/> +<s xml:id="echoid-s1664" xml:space="preserve"> +Viète used the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the whole line, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the first part to the first segment, +and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the second part to the second segment, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> for the prescribed difference, +and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> for the unknown parts of the first and second segments. +Harriot followed Viète's method on Add MS 6782, f. 464. +Here he works the same problem using numbers instead of lengths. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head214" xml:space="preserve" xml:lang="lat"> +Zetet. Lib. 1. Zet. 8. +<lb/>[<emph style="it">tr: +Zetetica, Book I, Zetetic 8 +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1666" xml:space="preserve"> +Datum numerum ita dividere in duas partes, ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> primæ partis, minus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> secundæ partis: +sit æqualis numero pæscripto. oportuit numerum præscriptum esse minorum quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> totius. +<lb/>[<emph style="it">tr: +To divide a given number into two parts, so that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> of the first part minus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of the second part +is equal to a prescribed number; the prescribed number must be less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> of the whole. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1667" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. numerus datus. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. prima pars <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. secunda pars. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> primæ partis. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. (minor quam 28) numerus præscriptus. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>h</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>. secundæ partis. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>a</mi><mo>-</mo><mi>h</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math> +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, the given number. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, the first part <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, the second part. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> of the first part. <lb/> +The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> (less than 28) is the prescribed number. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>h</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of the second part <lb/> +The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>a</mi><mo>-</mo><mi>h</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1668" xml:space="preserve"> +Secundò: <lb/> +Ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> primæ partis minus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> secundæ <lb/> +sit æqualis numero præscripto. oportet <lb/> +numerum præscriptum esse minorem quam <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> totius. +<lb/>[<emph style="it">tr: +Second. <lb/> +As <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of the first part minus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> of the second is equal to the prescribed number, +the prescribed number must be less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of the total. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1669" xml:space="preserve"> +Analogia solvens problema. +<lb/>[<emph style="it">tr: +A ratio for solving the problem. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1670" xml:space="preserve"> +Aliud exemplum. ubi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> sit 24. minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> et maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. <lb/> +<lb/>[<emph style="it">tr: +Another example, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is 24, less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> and greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1671" xml:space="preserve"> +Exemplum 2<emph style="super">i</emph> casus. <lb/> +sit iam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>. primæ partis. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>h</mi></mstyle></math> secundæ partis +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>a</mi><mo>-</mo><mi>h</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math> +<lb/>[<emph style="it">tr: +Example 2. <lb/> +Now let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be a <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of the first part. <lb/> +The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>h</mi></mstyle></math> is the second part. +The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>a</mi><mo>-</mo><mi>h</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math> +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f462v" o="462v" n="925"/> +<pb file="add_6782_f463" o="463" n="926"/> +<div xml:id="echoid-div304" type="page_commentary" level="2" n="304"> +<p> +<s xml:id="echoid-s1672" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1672" xml:space="preserve"> +This page contains Harriot's working of Zetetic 1 from Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book I. +</s> +<lb/> +<quote xml:lang="lat"> +Zeteticum I <lb/> +Data differentia duorum laterum, & adgregato eorumdem, invenire latera. +</quote> +<lb/> +<quote> +Given the difference of two roots, and their sum, find the roots. +</quote> +<lb/> +<s xml:id="echoid-s1673" xml:space="preserve"> +Viète used the letters <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> for the two roots, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for their difference, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> for their sum. +Harriot repeated Viète's working in his own symbolic notation. +In the lower half of the page, he refers to a proposition in Viète's +<emph style="it">Effectionum geometricarum</emph>, Proposition 12, +where a similar problem is solved geometrcially. +In Harriot's diagrams, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> represent the two extremes. +</s> +<lb/> +<quote xml:lang="lat"> +Propositio XII <lb/> +Data media trium proportionalium & differentia extremarum, invenire extremas. +</quote> +<lb/> +<s xml:id="echoid-s1674" xml:space="preserve"> +Given the mean of three proportionals, and the difference of their extremes, find the extremes. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head215" xml:space="preserve" xml:lang="lat"> +Zet. 1. lib. 1. +<lb/>[<emph style="it">tr: +Zetetic 1, Book I. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1676" xml:space="preserve"> +alia diagrapha. <lb/> +consule 12. p. effect. +<lb/>[<emph style="it">tr: +another diagram; see Effectionum, proposition 12. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f463v" o="463v" n="927"/> +<pb file="add_6782_f464" o="464" n="928"/> +<div xml:id="echoid-div305" type="page_commentary" level="2" n="305"> +<p> +<s xml:id="echoid-s1677" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1677" xml:space="preserve"> +This page contains Harriot's working of Zetetic 8 from Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book I. +</s> +<lb/> +<quote xml:lang="lat"> +Zeteticum VIII <lb/> +Datum latus ita secare, ut præfinitae unciae segmenti, multatæ præfinitis unciis secundi segmenti: +æquent differentiam praescriptam. +</quote> +<lb/> +<quote> +To cut a given line in such a way that a fixed part of one segment, subtracted from a fixed part of the second segment, +is equal to a prescribed difference. +</quote> +<lb/> +<s xml:id="echoid-s1678" xml:space="preserve"> +Viète used the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the whole line, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the first part to the first segment, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the ratio of the second part to the second segment, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> for the prescribed difference, +and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> for the parts of the first and second segments. +Harriot repeated Viète's working in his own notation, and also added some variants of his own. <lb/> +The work is continued using numerical examples on Add MS 6782, f. 462. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head216" xml:space="preserve" xml:lang="lat"> +Zetet. Lib. 1. Zet. 8. +<lb/>[<emph style="it">tr: +Zetetica, Book I, Zetetic 8 +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1680" xml:space="preserve"> +Datum latus ita secare, ut præfinitae unciæ segmenti, multatæ præfinitis unciis secundi segmenti: +æquent differentiam præscriptam. +<lb/>[<emph style="it">tr: +To cut a given line in such a way that a fixed part of one segment, subtracted from a fixed part of the second segment, +is equal to a prescribed difference. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1681" xml:space="preserve"> +Nota. <lb/> +Differentia præscripta <lb/> +videlicet <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> debet esse <lb/> +minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> <lb/> +sive <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> sit maior vel <lb/> +minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>, ut <lb/> +infra patebit. <lb/> +Hic sequens argu-<lb/> +mentatio <emph style="super">est</emph> firma ad <lb/> +utraque casum. +<lb/>[<emph style="it">tr: +Note. <lb/> +The prescribed difference, namely <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>, must be less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> +whether <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> is greater or less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>, as is shown below. +Here the following argument is sound in either case. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1682" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. latus secandam. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. primum segmentum <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. secundum seg. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. portio primæ seg. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math> <lb/> +Hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>a</mi><mo>,</mo><mi>o</mi></mstyle></math> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. differentia præscripta <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mfrac><mrow><mi>a</mi><mo>-</mo><mi>h</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math> <lb/> +Hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>a</mi><mo>-</mo><mi>h</mi><mo>,</mo><mi>u</mi></mstyle></math> <lb/> +Quæruntur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, et segmentorum portiones. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>a</mi><mo>,</mo><mfrac><mrow><mi>b</mi><mi>a</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math>. primum seg. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>a</mi><mo>-</mo><mi>h</mi><mo>,</mo><mfrac><mrow><mi>b</mi><mi>a</mi><mo>-</mo><mi>b</mi><mi>h</mi></mrow><mrow><mi>f</mi></mrow></mfrac></mstyle></math>. secundum seg. <lb/> +<lb/>[...]<lb/> <lb/> +Notum. Inde primum seg <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>b</mi><mi>a</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math>. +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, the line to be cut. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, the first segment <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, the second segment. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, the portion of the first segment. <lb/> +The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>o</mi></mrow></mfrac></mstyle></math> <lb/> +That is: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>a</mi><mo>:</mo><mi>o</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>, the prescribed difference <lb/> +The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>f</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>a</mi><mo>-</mo><mi>h</mi></mrow><mrow><mi>u</mi></mrow></mfrac></mstyle></math> <lb/> +That is: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>a</mi><mo>-</mo><mi>h</mi><mo>:</mo><mi>u</mi></mstyle></math> <lb/> +There are sought <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, and the portions of the segments. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>a</mi><mo>:</mo><mfrac><mrow><mi>b</mi><mi>a</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math>, the first segment <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>a</mi><mo>-</mo><mi>h</mi><mo>:</mo><mfrac><mrow><mi>b</mi><mi>a</mi><mo>-</mo><mi>b</mi><mi>h</mi></mrow><mrow><mi>f</mi></mrow></mfrac></mstyle></math>, the second segement. <lb/> +<lb/>[...]<lb/> <lb/> +Note. Thence the first segment, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>b</mi><mi>a</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1683" xml:space="preserve"> +Porrò, sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> portio <lb/> +secundi segmenti. +<lb/>[<emph style="it">tr: +Further, if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> is the portion of the second segment. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1684" xml:space="preserve"> +Additio nostra pro <lb/> +portione secundi segmenti, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> <lb/> +aliter. <lb/> +sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, portio secundi segmenti <lb/> +Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>+</mo><mi>h</mi></mstyle></math> portio primi segmenti. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>e</mi><mo>,</mo><mfrac><mrow><mi>b</mi><mi>e</mi></mrow><mrow><mi>f</mi></mrow></mfrac></mstyle></math>. secundum seg. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>e</mi><mo>+</mo><mi>h</mi><mo>,</mo><mfrac><mrow><mi>b</mi><mi>e</mi><mo>+</mo><mi>b</mi><mi>h</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math>. prim. seg. <lb/> +<lb/>[...]<lb/> <lb/> +Notum inde sec. segmentum: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>b</mi><mi>e</mi></mrow><mrow><mi>f</mi></mrow></mfrac></mstyle></math>. +<lb/>[<emph style="it">tr: +An addition of my own for the portion of the second segment, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, another way. <lb/> +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the portion of the second segment, therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>+</mo><mi>h</mi></mstyle></math> is the portion of the first segment. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>e</mi><mo>:</mo><mfrac><mrow><mi>b</mi><mi>e</mi></mrow><mrow><mi>f</mi></mrow></mfrac></mstyle></math>, the second segment. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>e</mi><mo>+</mo><mi>h</mi><mo>:</mo><mfrac><mrow><mi>b</mi><mi>e</mi><mo>+</mo><mi>b</mi><mi>h</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math>, the first segment. <lb/> +<lb/>[...]<lb/> <lb/> +Note, thence the second segment: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>b</mi><mi>e</mi></mrow><mrow><mi>f</mi></mrow></mfrac></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1685" xml:space="preserve"> +Additio nostra. Aliter 1<emph style="super">o</emph>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>,</mo><mi>d</mi><mo>:</mo><mi>o</mi><mo>,</mo><mfrac><mrow><mi>b</mi><mi>o</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math>. port. 1. seg. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>,</mo><mi>f</mi><mo>:</mo><mi>b</mi><mo>-</mo><mi>o</mi><mo>,</mo><mfrac><mrow><mi>f</mi><mi>b</mi><mo>-</mo><mi>f</mi><mi>o</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>. por. 2. seg. +<lb/>[<emph style="it">tr: +An addition of my own. Another way, 1. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>:</mo><mi>d</mi><mo>=</mo><mi>o</mi><mo>:</mo><mfrac><mrow><mi>b</mi><mi>o</mi></mrow><mrow><mi>d</mi></mrow></mfrac></mstyle></math>, the portion of the first segment. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>:</mo><mi>f</mi><mo>=</mo><mi>b</mi><mo>-</mo><mi>o</mi><mo>:</mo><mfrac><mrow><mi>f</mi><mi>b</mi><mo>-</mo><mi>f</mi><mi>o</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>, the portion of the second segment. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1686" xml:space="preserve"> +Aliter 2<emph style="super">o</emph>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>,</mo><mi>f</mi><mo>:</mo><mi>u</mi><mo>,</mo><mfrac><mrow><mi>f</mi><mi>u</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>. port. 2. seg. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>,</mo><mi>d</mi><mo>:</mo><mi>b</mi><mo>-</mo><mi>u</mi><mo>,</mo><mfrac><mrow><mi>d</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>u</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>. por. 1. seg. +<lb/>[<emph style="it">tr: +Another way, 2. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>:</mo><mi>f</mi><mo>=</mo><mi>u</mi><mo>:</mo><mfrac><mrow><mi>f</mi><mi>u</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>, the portion of the second segment. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>:</mo><mi>d</mi><mo>=</mo><mi>b</mi><mo>-</mo><mi>u</mi><mo>:</mo><mfrac><mrow><mi>d</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>u</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>, the portion of the first segment. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1687" xml:space="preserve"> +Nota. Hinc apparet quod <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> est minor <lb/> +quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>; alias <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> esset minor <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>; totum [???] partem. +<lb/>[<emph style="it">tr: +Note. Here it is clear that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>; +otherwise <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> would be less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>; the whole [???] the part. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f464v" o="464v" n="929"/> +<pb file="add_6782_f465" o="465" n="930"/> +<div xml:id="echoid-div306" type="page_commentary" level="2" n="306"> +<p> +<s xml:id="echoid-s1688" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1688" xml:space="preserve"> +This page contains Harriot's working of Zetetic 6 from Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book I. +</s> +<lb/> +<quote xml:lang="lat"> +Zeteticum VI <lb/> +Datis duobus lateribs uno deficiente à justo, altero justum excedente, una cum ratione defectus ad excessum: +invenire latus justum. +</quote> +<lb/> +<quote> +Given two roots, one less than the correct root, the other exceeding it, +together with the ratio of the defect to the excess, find the correct root. +</quote> +<lb/> +<s xml:id="echoid-s1689" xml:space="preserve"> +Viète used the letters <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> for the two roots, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>R</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>S</mi></mstyle></math> for the ratio of the defect to the excess. +Harriot repeated Viète's working, including his alternative method ('Aliter'), in his own symbolic notation. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head217" xml:space="preserve" xml:lang="lat"> +Zetet. lib. 1. Zet. 6. +<lb/>[<emph style="it">tr: +Zetetica, Book I, Zetetic 6. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1691" xml:space="preserve"> +Datis duobus lateribs uno deficiente à justo, altero justum excedente, una <lb/> +cum ratione defectus ad excessum: invenire latus justum. +<lb/>[<emph style="it">tr: +Given two roots, one less than the correct side, the other exceeding it, +together with the ratio of the deficiency to the excess, find the correct root. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1692" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. deficiens a justo. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. excedens justium. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>. defectus <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>. excessus ratione +primò. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. esto defectus a justo. <lb/> +Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>a</mi></mstyle></math>. latus iustum. <lb/> +Quoniam: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>,</mo><mi>s</mi><mo>:</mo><mi>a</mi><mo>,</mo><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mstyle></math> excessus <lb/> +<lb/>[...]<lb/> <lb/> +Ergo 80 latus iustum. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the deficiency from the correct root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> the excess over the correct root, +and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> the ratio of the defect to the excess. <lb/> +First, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be the defect from the correct side. <lb/> +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>a</mi></mstyle></math> is the correct root. <lb/> +Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>:</mo><mi>s</mi><mo>=</mo><mi>a</mi><mo>:</mo><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mstyle></math>, the excess. <lb/> +<lb/>[...]<lb/> <lb/> +Therefore 80 is the correct root. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1693" xml:space="preserve"> +Secundò <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. esto excessus. <lb/> +Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>e</mi></mstyle></math>. latus iustum. <lb/> +Tum: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mo>,</mo><mi>r</mi><mo>:</mo><mi>e</mi><mo>,</mo><mfrac><mrow><mi>r</mi><mi>e</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mstyle></math>. defectus. <lb/> +<lb/>[...]<lb/> <lb/> +Ergo 80 latus iustum. +<lb/>[<emph style="it">tr: +Second. <lb/> +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the excess. <lb/> +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>e</mi></mstyle></math> is the correct root. <lb/> +Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mo>:</mo><mi>r</mi><mo>=</mo><mi>e</mi><mo>:</mo><mfrac><mrow><mi>r</mi><mi>e</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mstyle></math>, the defect. <lb/> +<lb/>[...]<lb/> <lb/> +Therefore 80 is the correct root. +</emph>]<lb/> +</s> +</p> +<head xml:id="echoid-head218" xml:space="preserve" xml:lang="lat"> +Aliter +<lb/>[<emph style="it">tr: +Another way. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1694" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. esto latus iustum. <lb/> +Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>b</mi></mstyle></math>. defectus. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>a</mi></mstyle></math>. excessus. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be the correct root. <lb/> +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>a</mi></mstyle></math> is the defect, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>a</mi></mstyle></math> the excess. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f465v" o="465v" n="931"/> +<pb file="add_6782_f466" o="466" n="932"/> +<div xml:id="echoid-div307" type="page_commentary" level="2" n="307"> +<p> +<s xml:id="echoid-s1695" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1695" xml:space="preserve"> +This page contains Harriot's working of Zetetic 5 from Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book I. +</s> +<lb/> +<quote xml:lang="lat"> +Zeteticum V <lb/> +Datis duobus lateribus excedentibus justum, una cum ratione excessuum: invenire latus justum. +</quote> +<lb/> +<quote> +Given two roots exceeding the correct one, and the ratio of the excesses, find the correct root. +</quote> +<lb/> +<s xml:id="echoid-s1696" xml:space="preserve"> +Viète used the letters <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> for the two roots, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>R</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>S</mi></mstyle></math> for the ratio of the excesses. +Harriot repeated Viète's working, including his alternative method ('Aliter'), in his own symbolic notation. +For Harriot's (and Viète's) use of the symbol that looks like a modern = sign, +see the commentary to Add MS 6782, f. 467. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head219" xml:space="preserve" xml:lang="lat"> +Zetet. lib. 1. Zet. 5. +<lb/>[<emph style="it">tr: +Zetetica, Book I, Zetetic 5. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1698" xml:space="preserve"> +Datis duobus lateribus excedentibus iustum, una cum ratione excessum: <lb/> +invenire latus iustum. +<lb/>[<emph style="it">tr: +Given two roots exceeding the correct root, and the ratio of the excesses, find the correct root. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1699" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. latus primum. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. latus secundum. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>. excessus primi <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>. excessus secundi ratione +primò. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. esto excessus primi. <lb/> +Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>a</mi></mstyle></math>. latus iustum. <lb/> +Tum: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>,</mo><mi>s</mi><mo>:</mo><mi>a</mi><mo>,</mo><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mstyle></math> Excessus 2<emph style="super">i</emph> <lb/> +<lb/>[...]<lb/> <lb/> +Ergo latus iustum 20. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the first side, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> the second, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> the ratio of the excesses. <lb/> +First, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be the first excess. <lb/> +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>a</mi></mstyle></math> is the correct root. <lb/> +Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>:</mo><mi>s</mi><mo>=</mo><mi>a</mi><mo>:</mo><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mstyle></math>, the second excess. <lb/> +<lb/>[...]<lb/> <lb/> +Therefore the correct root is 20. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1700" xml:space="preserve"> +Secundò <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. esto excessus secundi. <lb/> +Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>e</mi></mstyle></math>. latus iustum. <lb/> +Tum: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mo>,</mo><mi>r</mi><mo>:</mo><mi>e</mi><mo>,</mo><mfrac><mrow><mi>r</mi><mi>e</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mstyle></math>. excessus primi. <lb/> +<lb/>[...]<lb/> <lb/> +Ergo latus iustum 30. +<lb/>[<emph style="it">tr: +Second. <lb/> +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the second excess. <lb/> +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>e</mi></mstyle></math> is the correct root. <lb/> +Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mo>:</mo><mi>r</mi><mo>=</mo><mi>e</mi><mo>:</mo><mfrac><mrow><mi>r</mi><mi>e</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mstyle></math>, the first excess. <lb/> +<lb/>[...]<lb/> <lb/> +Therefore the correct root is 30. +</emph>]<lb/> +</s> +</p> +<head xml:id="echoid-head220" xml:space="preserve" xml:lang="lat"> +Aliter +<lb/>[<emph style="it">tr: +Another way. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1701" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. esto latus iustum. <lb/> +Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>a</mi></mstyle></math>. excessus primi. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>a</mi></mstyle></math>. excessus secundi. +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be the correct root. <lb/> +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>a</mi></mstyle></math> is the first excess, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>-</mo><mi>a</mi></mstyle></math> the second excess. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1702" xml:space="preserve"> +20. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. latus iustum +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn><mn>0</mn></mstyle></math>, the correct root. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1703" xml:space="preserve"> +Nota. <lb/> +Si notatio ita fiat. sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. latus minus <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. latus maius <lb/> +&c. <lb/> +Tunc non opus esset isto <lb/> +signo =. +<lb/>[<emph style="it">tr: +Note. <lb/> +Of the notation is thus: let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the smaller side, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> the larger, etc. <lb/> +Then there would be no need for this sign =. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f466v" o="466v" n="933"/> +<pb file="add_6782_f467" o="467" n="934"/> +<div xml:id="echoid-div308" type="page_commentary" level="2" n="308"> +<p> +<s xml:id="echoid-s1704" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1704" xml:space="preserve"> +This page contains Harriot's working of Zetetic 4 from Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book I. +</s> +<lb/> +<quote xml:lang="lat"> +Zeteticum IV <lb/> +Datis duobus lateribus deficientibus à justo, una cum ratione defectum: invenire latus justum. +</quote> +<lb/> +<quote> +Given two roots less than the correct one, and the ratio of the defects, find the correct root. +</quote> +<lb/> +<s xml:id="echoid-s1705" xml:space="preserve"> +Viète used the letters <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> for the two roots, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>R</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>S</mi></mstyle></math> for the ratio of the defects. +Harriot repeated Viète's working, including his alternative method ('Aliter'),in his own symbolic notation. +The symbol that looks like a modern = sign is to be read as a minus sign, +used by Harriot (following Viète) in cases where it was not known which quantity was greater. +Thus, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math> is to be read as '<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>b</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>a</mi></mstyle></math>, whichever is positive'. +In modern notation the same result is represented by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo lspace="0em" rspace="0em" maxsize="1">|</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo lspace="0em" rspace="0em" maxsize="1">|</mo></mstyle></math>. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head221" xml:space="preserve" xml:lang="lat"> +Zetet. lib. 1. Zet. 4. +<lb/>[<emph style="it">tr: +Zetetica, Book I, Zetetic 4 +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1707" xml:space="preserve"> +Datis duobus lateribus deficientibus a iusto, una cum ratione defectum: <lb/> +invenire latus iustum. +<lb/>[<emph style="it">tr: +Given two roots less than the true one, together with the ratio of the defects, find the true root. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1708" xml:space="preserve"> +primò. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. latus primum. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. latus secundum. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>. defectus primum <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>. defectus secundi ratione +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. esto defectus primi. <lb/> +Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>a</mi></mstyle></math>. latus iustum. <lb/> +Tum: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>,</mo><mi>s</mi><mo>:</mo><mi>a</mi><mo>,</mo><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mstyle></math> +defect. 2<emph style="super">i</emph> <lb/> +<lb/>[...]<lb/> <lb/> +Ergo latus iustum: 60. +<lb/>[<emph style="it">tr: +First. <lb/> +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the first root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> the second, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> the ratio of the first defect to the second, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> the first defect. <lb/> +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>a</mi></mstyle></math> is the true root. <lb/> +Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>:</mo><mi>s</mi><mo>=</mo><mi>a</mi><mo>:</mo><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mstyle></math>, the second defect. <lb/> +<lb/>[...]<lb/> <lb/> +Therefore the correct root is 60. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1709" xml:space="preserve"> +secundò <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. esto defectus secundi. <lb/> +Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>+</mo><mi>e</mi></mstyle></math>. latus iustum. <lb/> +Tum: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mo>,</mo><mi>r</mi><mo>:</mo><mi>e</mi><mo>,</mo><mfrac><mrow><mi>r</mi><mi>e</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mstyle></math>. <lb/> +<lb/>[...]<lb/> <lb/> +Ergo latus iustum: 60. +<lb/>[<emph style="it">tr: +Second <lb/> +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the second defect. <lb/> +Therefore the true root is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>+</mo><mi>e</mi></mstyle></math>. <lb/> +Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mo>:</mo><mi>r</mi><mo>=</mo><mi>e</mi><mo>:</mo><mfrac><mrow><mi>r</mi><mi>e</mi></mrow><mrow><mi>s</mi></mrow></mfrac></mstyle></math>. <lb/> +<lb/>[...]<lb/> <lb/> +Therefore the correct root is 60. +</emph>]<lb/> +</s> +</p> +<head xml:id="echoid-head222" xml:space="preserve" xml:lang="lat"> +Aliter +<lb/>[<emph style="it">tr: +Another way. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1710" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. esto latus iustum. <lb/> +Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>b</mi></mstyle></math>. defectus primi. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>d</mi></mstyle></math>. defectus secundi. <lb/> +Quære: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>b</mi><mo>,</mo><mi>a</mi><mo>-</mo><mi>d</mi><mo>:</mo><mi>r</mi><mo>,</mo><mi>s</mi></mstyle></math>. <lb/> +Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>a</mi><mo>-</mo><mi>r</mi><mi>d</mi><mo>=</mo><mi>s</mi><mi>a</mi><mo>-</mo><mi>s</mi><mi>b</mi></mstyle></math> <lb/> +Hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mi>b</mi><mo>=</mo><mo>=</mo><mi>r</mi><mi>d</mi><mo>=</mo><mi>s</mi><mi>a</mi><mo>=</mo><mo>=</mo><mi>r</mi><mi>a</mi></mstyle></math>. <lb/> +Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>s</mi><mi>b</mi><mo>=</mo><mo>=</mo><mi>r</mi><mi>d</mi></mrow><mrow><mi>s</mi><mo>=</mo><mo>=</mo><mi>r</mi></mrow></mfrac><mo>=</mo><mi>a</mi></mstyle></math> +<lb/>[<emph style="it">tr: +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be the correct root. <lb/> +Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>b</mi></mstyle></math> is the first defect, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>d</mi></mstyle></math> the second. <lb/> +Obtain: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>b</mi><mo>:</mo><mi>a</mi><mo>-</mo><mi>d</mi><mo>=</mo><mi>r</mi><mo>:</mo><mi>s</mi></mstyle></math>. <lb/> +Therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo lspace="0em" rspace="0em" maxsize="1">|</mo><mi>r</mi><mi>a</mi><mo>-</mo><mi>r</mi><mi>d</mi><mo lspace="0em" rspace="0em" maxsize="1">|</mo><mo>=</mo><mo lspace="0em" rspace="0em" maxsize="1">|</mo><mi>s</mi><mi>a</mi><mo>-</mo><mi>s</mi><mi>b</mi><mo lspace="0em" rspace="0em" maxsize="1">|</mo></mstyle></math>. <lb/> +That is: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mo lspace="0em" rspace="0em" maxsize="1">|</mo><mi>s</mi><mi>b</mi><mo>-</mo><mi>r</mi><mi>d</mi><mo lspace="0em" rspace="0em" maxsize="1">|</mo></mrow><mrow><mo lspace="0em" rspace="0em" maxsize="1">|</mo><mi>s</mi><mo>-</mo><mi>r</mi><mo lspace="0em" rspace="0em" maxsize="1">|</mo></mrow></mfrac><mo>=</mo><mi>a</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1711" xml:space="preserve"> +<emph style="st"> +Emendata Vieta <lb/> +si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> sit minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>, minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>a</mi><mo>-</mo><mi>r</mi><mi>d</mi><mo>=</mo><mi>s</mi><mi>a</mi><mo>-</mo><mi>s</mi><mi>b</mi></mstyle></math>. <lb/> +Tum si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> sit maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>: vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>. <lb/> +Inde: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>a</mi><mo>-</mo><mi>s</mi><mi>a</mi><mo>=</mo><mi>r</mi><mi>d</mi><mo>-</mo><mi>s</mi><mi>b</mi></mstyle></math> <lb/> +Vel: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>e</mi><mo>+</mo><mi>s</mi><mi>e</mi><mo>=</mo><mi>s</mi><mi>g</mi></mstyle></math>. <lb/> +Unde: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mfrac><mrow><mi>r</mi><mi>d</mi><mo>-</mo><mi>s</mi><mi>b</mi></mrow><mrow><mi>r</mi><mo>-</mo><mi>s</mi></mrow></mfrac></mstyle></math> +<lb/>[<emph style="it">tr: +A correction to Viète. <lb/> +If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is smaller than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> is smaller than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>a</mi><mo>-</mo><mi>r</mi><mi>d</mi><mo>-</mo><mi>s</mi><mi>a</mi><mo>-</mo><mi>s</mi><mi>b</mi></mstyle></math>. <lb/> +Then if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, +thence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>a</mi><mo>-</mo><mi>s</mi><mi>a</mi><mo>-</mo><mi>r</mi><mi>d</mi><mo>-</mo><mi>s</mi><mi>b</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>e</mi><mo>+</mo><mi>s</mi><mi>e</mi><mo>=</mo><mi>s</mi><mi>g</mi></mstyle></math>. Whence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mfrac><mrow><mi>r</mi><mi>d</mi><mo>-</mo><mi>s</mi><mi>b</mi></mrow><mrow><mi>r</mi><mo>-</mo><mi>s</mi></mrow></mfrac></mstyle></math>. +</emph>]<lb/> +</emph> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1712" xml:space="preserve"> +Nota <lb/> +Si notatis ita fiat: sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. latus minus. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> latus maius. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> defectus minus <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> defectus maius ratione <lb/> +Tunc non opus est isto signo = +<lb/>[<emph style="it">tr: +Note <lb/> +If the notation is thus: let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the smaller root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> the larger root, +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> the ratio of the smaller defect to the larger, then there is no need for this sign == +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f467v" o="467v" n="935"/> +<div xml:id="echoid-div309" type="page_commentary" level="2" n="309"> +<p> +<s xml:id="echoid-s1713" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1713" xml:space="preserve"> +This page contains some of Harriot's rough working on Zetetic 9 from Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book I. <lb/> +For fuller treatments of the material see Add MS 6782, f. 460 adn f. 461. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head223" xml:space="preserve" xml:lang="lat"> +Zet. 9. Cas. 2 +<lb/>[<emph style="it">tr: +Zetetic 9, case 2. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1715" xml:space="preserve"> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. portio primi lateris et <emph style="st">maioris</emph> minimis. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math>. portio secundi et maioris <lb/> +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, the portion of the first line, the lesser. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>-</mo><mi>a</mi></mstyle></math>, the portion of the second and greater line. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1716" xml:space="preserve"> +sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. portio secundi lateris et maioris. +<lb/>[<emph style="it">tr: +et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the portion of the second and greater line. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1717" xml:space="preserve"> +Primum latus minus. Portio minorum uncium. +<lb/>[<emph style="it">tr: +The first line smaller. Portion less than the fraction. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1718" xml:space="preserve"> +Primum latus maius. Portio minorum uncium. +<lb/>[<emph style="it">tr: +The first line greater. Portion less than the fraction. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1719" xml:space="preserve"> +Primum latus minus. +<lb/>[<emph style="it">tr: +The first line smaller. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f468" o="468" n="936"/> +<div xml:id="echoid-div310" type="page_commentary" level="2" n="310"> +<p> +<s xml:id="echoid-s1720" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1720" xml:space="preserve"> +The reference here is to Viète, +<emph style="it">Ad problema, quod ... proposuit Adrianus Romanus, responsum</emph> (1595), +pages 5, 7, and 13. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head224" xml:space="preserve" xml:lang="lat"> +ad pag: 5.7. & 13. Vietæ ad Adrianum. videlicet responsi +<lb/>[<emph style="it">tr: +On pages 5, 7, and 13 of Viète, Adrianus, that is, his responses. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1722" xml:space="preserve"> +Aliter. +<lb/>[<emph style="it">tr: +Another way. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1723" xml:space="preserve"> +Excedit semicirculum +<lb/>[<emph style="it">tr: +Exceeds a semicircle +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f468v" o="468v" n="937"/> +<pb file="add_6782_f469" o="469" n="938"/> +<div xml:id="echoid-div311" type="page_commentary" level="2" n="311"> +<p> +<s xml:id="echoid-s1724" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1724" xml:space="preserve"> +The positions of this page amongst other pages referring to Viète's +<emph style="it">Adrianus Romanus responsum</emph>, and its subject matter, +trisection and quinquisection of an angle, suggests that it relates to +<emph style="it">Adrianus Romanus responsum</emph>, Chapter V. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s1726" xml:space="preserve"> +Pentachotomia +<lb/>[<emph style="it">tr: +Quinquisection of an angle +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1727" xml:space="preserve"> +Trichotomia +<lb/>[<emph style="it">tr: +Trisection of an angle +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f469v" o="469v" n="939"/> +<div xml:id="echoid-div312" type="page_commentary" level="2" n="312"> +<p> +<s xml:id="echoid-s1728" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1728" xml:space="preserve"> +The positions of this page amongst other pages referring to Viète's +<emph style="it">Adrianus Romanus responsum</emph>, and its subject matter, +trisection and quinquisection of an angle, suggests that it relates to +<emph style="it">Adrianus Romanus responsum</emph>, Chapter V. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s1730" xml:space="preserve"> +Pentachotomia +<lb/>[<emph style="it">tr: +Quinquisection of an angle +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f470" o="470" n="940"/> +<div xml:id="echoid-div313" type="page_commentary" level="2" n="313"> +<p> +<s xml:id="echoid-s1731" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1731" xml:space="preserve"> +The references on this page are to Viète, +<emph style="it">Ad problema, quod ... proposuit Adrianus Romanus, responsum</emph> (1595), +pages 7v and 5. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head225" xml:space="preserve" xml:lang="lat"> +ad: pag: 7.b. +<lb/>[<emph style="it">tr: +On page 7v. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1733" xml:space="preserve"> +omisit Vieta radices negativas <lb/> +ut in omnibus alijs aequationibus +<lb/>[<emph style="it">tr: +Viète omits negative roots, as in all other equations. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1734" xml:space="preserve"> +Exemplum pag: 5. pono sub meliori forma <lb/> +quam Vieta, ut sequitur: <lb/> +et est generalis ad omnes sectiones anguli <lb/> +imparis numeri. +<lb/>[<emph style="it">tr: +Example from page 5, I put in a better form than Viète, as follows; +and it is general for all angular sections of odd number. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f470v" o="470v" n="941"/> +<div xml:id="echoid-div314" type="page_commentary" level="2" n="314"> +<p> +<s xml:id="echoid-s1735" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1735" xml:space="preserve"> +A continuation from Add MS 6782, f. 470, of work on Viète's, +<emph style="it">Ad problema, quod ... proposuit Adrianus Romanus, responsum</emph> (1595). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s1737" xml:space="preserve"> +aliter: sed melius in alia charta +<lb/>[<emph style="it">tr: +another way, but better in the other sheet +</emph>]<lb/> +[<emph style="it">Note: +The other sheet referred to here is the reverse of this one, Add MS 6782, f. 470. + </emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1738" xml:space="preserve"> +Trichotomia. sic brevie +<lb/>[<emph style="it">tr: +Trisection, thus briefly +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1739" xml:space="preserve"> +hinc sequitur: +<lb/>[<emph style="it">tr: +this follows: +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f471" o="471" n="942"/> +<div xml:id="echoid-div315" type="page_commentary" level="2" n="315"> +<p> +<s xml:id="echoid-s1740" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1740" xml:space="preserve"> +The positions of this page amongst other pages referring to Viète's +<emph style="it">Adrianus Romanus responsum</emph>, and its subject matter, +trisection of an angle, suggests that it relates to +<emph style="it">Adrianus Romanus responsum</emph>, Chapter V. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head226" xml:space="preserve" xml:lang="lat"> +Trichotomia +<lb/>[<emph style="it">tr: +Trisection +</emph>]<lb/> +</head> +<pb file="add_6782_f471v" o="471v" n="943"/> +<pb file="add_6782_f472" o="472" n="944"/> +<div xml:id="echoid-div316" type="page_commentary" level="2" n="316"> +<p> +<s xml:id="echoid-s1742" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1742" xml:space="preserve"> +This page contains further work on Zetetic 2 from Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book II. +</s> +<lb/> +<quote xml:lang="lat"> +Zeteticum II <lb/> +Dato rectangulo sub lateribus, & adgregato quadratorum, inveniuntur latera. +</quote> +<lb/> +<quote> +Given the product of two sides, and the sum of their squares, the sides may be found. +</quote> +<lb/> +<s xml:id="echoid-s1743" xml:space="preserve"> +Harriot began the problem on Add MS 6782, f. 472, but he described the treatment on this page as better. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head227" xml:space="preserve" xml:lang="lat"> +lib. 2. Zetet. Zet. 2. secundo et melius. +<lb/>[<emph style="it">tr: +Zetetica, Book II, Zetetic 2. Second way, and better. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1745" xml:space="preserve"> +Dato rectangulo sub lateribus et adgregato quadratorum, inveniuntur latera. <lb/> +Enimvero duplum planum sub lateribus adiectum quidem <lb/> +adgregatum quadratorum, æquatur quadrato summa laterum:] <lb/> +sit planum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>δ</mi></mstyle></math>, cui æquale <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>θ</mi></mstyle></math>. et adgregatum <lb/> +quadratorum laterum sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>γ</mi><mi>H</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>λ</mi></mstyle></math>. <lb/> +<lb/>[...]<lb/> <lb/> +Ergo datur summa latera. +<lb/>[<emph style="it">tr: +Given a rectangle from its sides, and the sum of their squares, there may be found the sides. <lb/> +Certainly, twic the rectangle from the sides added to the sum of the squares +is equal to the square of the sum of the sides. <lb/> +Let the plane be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>δ</mi></mstyle></math>, which is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>θ</mi></mstyle></math>, +and the sum of the squares of the sides is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>γ</mi><mi>H</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>λ</mi></mstyle></math>. <lb/> +<lb/>[...]<lb/> <lb/> +Therefore the sum of the sides is given. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1746" xml:space="preserve"> +Ablatum uno, quadrato differentiæ.] <lb/> +<lb/>[...]<lb/> <lb/> +ergo, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>\</mo><mi>a</mi><mi>l</mi><mi>p</mi><mi>h</mi><mi>μ</mi></mstyle></math> est differentia laterum <lb/> +et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ρ</mi><mi>o</mi></mstyle></math> est quadratum differenti +<lb/>[<emph style="it">tr: +Subtracting from one, the square of the difference, <lb/> +<lb/>[...]<lb/> +therefore, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>μ</mi></mstyle></math> is the difference of the sides, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ρ</mi><mi>o</mi></mstyle></math> is the square of the difference. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1747" xml:space="preserve"> +In notis: sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>δ</mi></mstyle></math> planum æquale <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math> . <lb/> +Ergo, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>+</mo><mn>2</mn><mo>,</mo><mi>c</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>a</mi></mstyle></math>. adgregati <lb/> +Et. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>-</mo><mn>2</mn><mo>,</mo><mi>c</mi><mi>c</mi><mo>=</mo><mi>e</mi><mi>e</mi></mstyle></math>. differentia <lb/> +Ergo per 1.z et primi lib. dantur latera <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>β</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>δ</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +In letters, let the plane <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>δ</mi></mstyle></math> be equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math>. <lb/> +Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>+</mo><mn>2</mn><mo>,</mo><mi>c</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>a</mi></mstyle></math>, the sum. <lb/> +And <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>-</mo><mn>2</mn><mo>,</mo><mi>c</mi><mi>c</mi><mo>=</mo><mi>e</mi><mi>e</mi></mstyle></math>, the difference <lb/> +Therefore, by Zetetic 1 from the first book, there are given the sides <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>β</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>δ</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1748" xml:space="preserve"> +Aliter. <lb/> +Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>x</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. et sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>δ</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Another way. <lb/> +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>x</mi></mstyle></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>δ</mi></mstyle></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f472v" o="472v" n="945"/> +<pb file="add_6782_f473" o="473" n="946"/> +<div xml:id="echoid-div317" type="page_commentary" level="2" n="317"> +<p> +<s xml:id="echoid-s1749" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1749" xml:space="preserve"> +This page contains some very brief observations on Zetetic 3 from Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book II. +</s> +<lb/> +<quote xml:lang="lat"> +Zeteticum III <lb/> +Dato rectangulo sub lateribus, & differentia laterum: inveniuntur latera. +</quote> +<lb/> +<quote> +Given the product of two sides and their difference, the sides may be found. +</quote> +<lb/> +<s xml:id="echoid-s1750" xml:space="preserve"> +This is Proposition I.30 from the <emph style="it">Arithmetica</emph> of Diophantus, +but Harriot refers only to Viète's version of it. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head228" xml:space="preserve" xml:lang="lat"> +lib. 2. Zet. <lb/> +Zet. 3. +<lb/>[<emph style="it">tr: +Zetetica, Book II, Zetetic 3. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1752" xml:space="preserve"> +Dato rectangulo sub lateribus, et differentia laterum inveniuntur latera. +<lb/>[<emph style="it">tr: +Given the product of two sides and their difference, the sides may be found. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1753" xml:space="preserve"> +Enimvero: +<lb/>[<emph style="it">tr: +Certainly: +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1754" xml:space="preserve"> +Sint notæ et in 2, zetetico. +<lb/>[<emph style="it">tr: +This is noted also in Zetetic 2. +</emph>]<lb/> +[<emph style="it">Note: +For Harriot's treatment of Zetetic 2, see Add MS 6782, f. 474 and f. 472. + </emph>]<lb/> +</s> +</p> +<pb file="add_6782_f473v" o="473v" n="947"/> +<pb file="add_6782_f474" o="474" n="948"/> +<div xml:id="echoid-div318" type="page_commentary" level="2" n="318"> +<p> +<s xml:id="echoid-s1755" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1755" xml:space="preserve"> +This page contains Harriot's working of Zetetic 2 from Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book II. +</s> +<lb/> +<quote xml:lang="lat"> +Zeteticum II <lb/> +Dato rectangulo sub lateribus, & adgregato quadratorum, inveniuntur latera. +</quote> +<lb/> +<quote> +Given the product of two sides, and the sum of their squares, the sides may be found. +</quote> +<lb/> +<s xml:id="echoid-s1756" xml:space="preserve"> +Harriot's treatment of the problem is continued on Add MS 6782, f. 472. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head229" xml:space="preserve" xml:lang="lat"> +lib. 2. Zeteticorum <lb/> +Zet. 2. primo +<lb/>[<emph style="it">tr: +Zetetica, Book II <lb/> +Zetetic 2. First way. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1758" xml:space="preserve"> +Dato rectangulo sub lateribus et adgregato quadratorum, inveniuntur latera. <lb/> +Hoc est: <lb/> +Data recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>γ</mi></mstyle></math> et plano <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>δ</mi></mstyle></math>; invenientur <lb/> +latera <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>δ</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>γ</mi></mstyle></math>. <lb/> +vel: <lb/> +Data recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>δ</mi></mstyle></math> et quadrato <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ε</mi></mstyle></math>: invenientur <lb/> +duæ <emph style="super">rectæ</emph> lineæ quæ cum data <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>γ</mi></mstyle></math>, +efficiunt triangulum rectangulum æquale dimidio <lb/> +dati quadrati <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ε</mi></mstyle></math>, ita ut linea data sit Hyponetusu. <lb/> +sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>δ</mi></mstyle></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>: Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>γ</mi><mi>δ</mi></mstyle></math> erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>c</mi><mi>c</mi></mrow><mrow><mi>a</mi></mrow></mfrac></mstyle></math>. +<lb/>[<emph style="it">tr: +Given a rectangle from its sides, and the sum of their squares, there may be found the sides. <lb/> +That is: <lb/> +Given the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>γ</mi></mstyle></math> and the surface <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>δ</mi></mstyle></math>, +there may be found the sides <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>δ</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>γ</mi></mstyle></math>. <lb/> +or: <lb/> +Given the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>γ</mi></mstyle></math> and the square <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ε</mi></mstyle></math>, +there may be found two straight lines which, with the given line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>γ</mi></mstyle></math>, +form a right-angled triangle equal to half the given square <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>ε</mi></mstyle></math>, +in such a way that the given line is the hypotenuse. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1759" xml:space="preserve"> +Aliter. <lb/> +Sit rectangulum datum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>δ</mi></mstyle></math>. cui æquale <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>θ</mi></mstyle></math>. <lb/> +et adgregatum quadratorum sit: <lb/> +<lb/>[...]<lb/> <lb/> +data igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>x</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Another way. <lb/> +Let the given rectangle be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>δ</mi></mstyle></math>, which is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>θ</mi></mstyle></math>, +and the sum of the squares is: <lb/> +<lb/>[...]<lb/> <lb/> +therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>x</mi></mstyle></math> is given. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1760" xml:space="preserve"> +Itaque in notis: sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>x</mi></mstyle></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> . <lb/> +<lb/>[...]<lb/> <lb/> +Inveniatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, et dicatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Thus in symbols: let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>x</mi></mstyle></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/> +<lb/>[...]<lb/> <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> is found, and is said to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1761" xml:space="preserve"> +Et <emph style="super">quia</emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, est minor <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +And because <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f474v" o="474v" n="949"/> +<pb file="add_6782_f475" o="475" n="950"/> +<div xml:id="echoid-div319" type="page_commentary" level="2" n="319"> +<p> +<s xml:id="echoid-s1762" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1762" xml:space="preserve"> +The reference at the top of this page is to Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book V, Zetetic 14. +</s> +<lb/> +<quote xml:lang="lat"> +Zeteticum XIV <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> quadratum minus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> plano adaequare uni quadrato, quod fit minus quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>, +sed majus quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>. +</quote> +<lb/> +<quote> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>-squared minus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> is equal to a square, which is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>A</mi></mstyle></math> but greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>A</mi></mstyle></math>. +</quote> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head230" xml:space="preserve" xml:lang="lat"> +Zeteticorum. lib. 5. Zet: 14. +<lb/>[<emph style="it">tr: +Zetetica, Book V, Zetetic 14. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1764" xml:space="preserve"> +minus quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>a</mi></mstyle></math> +<lb/>[<emph style="it">tr: +less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>a</mi></mstyle></math> +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1765" xml:space="preserve"> +maius quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math> +<lb/>[<emph style="it">tr: +greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math> +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1766" xml:space="preserve"> +Quæritur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, ponatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> is sought, suppose it is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1767" xml:space="preserve"> +solutio problematis +<lb/>[<emph style="it">tr: +solution to the problem +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f475v" o="475v" n="951"/> +<pb file="add_6782_f476" o="476" n="952"/> +<div xml:id="echoid-div320" type="page_commentary" level="2" n="320"> +<p> +<s xml:id="echoid-s1768" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1768" xml:space="preserve"> +The reference at the top of this page appears to be to Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book V, Zetetic 14, +as on Add MS 6782, f. 475. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head231" xml:space="preserve" xml:lang="lat"> +Zet. lib. Zet: 14. +<lb/>[<emph style="it">tr: +Zetetica, Book [V], Zetetic 14. +</emph>]<lb/> +</head> +<pb file="add_6782_f476v" o="476v" n="953"/> +<pb file="add_6782_f477" o="477" n="954"/> +<div xml:id="echoid-div321" type="page_commentary" level="2" n="321"> +<p> +<s xml:id="echoid-s1770" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1770" xml:space="preserve"> +This page contains a lemma needed for Zetetic 10 from Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book II. +</s> +<lb/> +<quote xml:lang="lat"> +Zeteticum X <lb/> +Dato plano, quod constat tum rectangulo sub lateribus, tum quadratis singulorum laterum, +datoque è lateribus uno, invenire latus reliquum.</quote> +<lb/> +<quote> +Given a plane, consisting of a rectangle from two sides together with the individual squares of the sides, +and given also one of the sides, find the remaining side. +</quote> +<lb/> +<s xml:id="echoid-s1771" xml:space="preserve"> +Viète called the given side <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>, and the side to be found <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>D</mi></mstyle></math>, +and arrived at the following conclusion: +</s> +<lb/> +<quote xml:lang=""> +Planum constans rectangulo sub lateribus, & quadratis singulorum laterum, +multatum dodrante quadrati lateris dati, æquale est quadrato lateris compositi, ex quæsito latere & dimido dati. +</quote> +<lb/> +<quote> +The plane consisting of a rectangle from the sides and the square of the individual sides, +minus three-quarters of the square of the given root, +is equal to the square of the line composed of the sought line and half the given line. +</quote> +<lb/> +<s xml:id="echoid-s1772" xml:space="preserve"> +Borrowing Viète's notation, his statement may be written as the identity +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>A</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>D</mi><mrow><msup><mo maxsize="1">)</mo><mn>2</mn></msup></mrow><mo>+</mo><mrow><msup><mi>D</mi><mn>2</mn></msup></mrow><mo>+</mo><mo maxsize="1">(</mo><mi>D</mi><mi>A</mi><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mrow><msup><mi>D</mi><mn>2</mn></msup></mrow><mo maxsize="1">)</mo><mo>=</mo><mrow><msup><mi>A</mi><mn>2</mn></msup></mrow></mstyle></math>. +On this page Harriot investigated the problem geometrically. +On Add MS 6782, f. 479v, he treated it algebraically. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head232" xml:space="preserve" xml:lang="lat"> +Lemma. ad Zet. 2, 10. +<lb/>[<emph style="it">tr: +Lemma to Zetetic II.10 +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1774" xml:space="preserve"> +Si recta linea <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>β</mi></mstyle></math> dividatur bisariam in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>γ</mi></mstyle></math>: et ei adjiciatur altera <lb/> +linea <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>δ</mi></mstyle></math>: dico quod: +<lb/>[<emph style="it">tr: +If the straight line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>β</mi></mstyle></math> is divided in two at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>γ</mi></mstyle></math>, +and to it is added another line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>δ</mi></mstyle></math>, I say that: +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1775" xml:space="preserve"> +Aliter. <lb/> +Sit rectangulum datum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>δ</mi></mstyle></math>. cui æquale <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>θ</mi></mstyle></math>. <lb/> +et adgregatum quadratorum sit: <lb/> +<lb/>[...]<lb/> <lb/> +data igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>x</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Another way. <lb/> +Let the given rectangle be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>δ</mi></mstyle></math>, which is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>θ</mi></mstyle></math>, +and the sum of the squares is: <lb/> +<lb/>[...]<lb/> <lb/> +therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>x</mi></mstyle></math> is given. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f477v" o="477v" n="955"/> +<pb file="add_6782_f478" o="478" n="956"/> +<pb file="add_6782_f478v" o="478v" n="957"/> +<pb file="add_6782_f479" o="479" n="958"/> +<pb file="add_6782_f479v" o="479v" n="959"/> +<div xml:id="echoid-div322" type="page_commentary" level="2" n="322"> +<p> +<s xml:id="echoid-s1776" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1776" xml:space="preserve"> +There are margin notes on this page referring to Zetetica 9 and 10 from Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book II. +</s> +<lb/> +<quote xml:lang=""> +Zeteticum IX <lb/> +Dato rectangulo sub lateribus, & differentia quadratorum, invenire latera. +</quote> +<lb/> +<quote> +Given a rectangle from two sides and the difference of their squares, find the sides. +</quote> +<lb/> +<quote xml:lang="lat"> +Zeteticum X <lb/> +Dato plano, quod constat tum rectangulo sub lateribus, tum quadratis singulorum laterum, +datoque è lateribus uno, invenire latus reliquum.</quote> +<lb/> +<quote> +Given a plane, consisting of a rectangle from two sides together with the individual squares of the sides, +and given also one of the sides, find the remaining side. +</quote> +<lb/> +<s xml:id="echoid-s1777" xml:space="preserve"> +For a fuller investigation of Zetetic X, see Add MS 6782, f. 477. +</s> +<lb/> +<s xml:id="echoid-s1778" xml:space="preserve"> +At the bottom of the page there is what appears to be a list of books. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<p xml:lang="lat"> +<s xml:id="echoid-s1780" xml:space="preserve"> +Zet.2,9 +<lb/>[<emph style="it">tr: +Zetetic II.9 +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1781" xml:space="preserve"> +cubuc rectanguli sub <lb/> +lateribus +<lb/>[<emph style="it">tr: +cube of the rectangle from the sides +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1782" xml:space="preserve"> +Zet.2,10 +<lb/>[<emph style="it">tr: +Zetetic II.10 +</emph>]<lb/> +</s> +</p> +<p> +<s xml:id="echoid-s1783" xml:space="preserve"> +[???] <lb/> +[???] <lb/> +[???] <lb/> +[???] <lb/> +A bagg of books [???] <lb/> +Vitello <lb/> +Euclid. 2. vol. <lb/> +physica Arist. [???] <lb/> +Ethica Piccolomini +</s> +</p> +<pb file="add_6782_f480" o="480" n="960"/> +<div xml:id="echoid-div323" type="page_commentary" level="2" n="323"> +<p> +<s xml:id="echoid-s1784" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1784" xml:space="preserve"> +The reference at the top of this page is to Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book V, Zetetic 4. +</s> +<lb/> +<quote xml:lang="lat"> +Zeteticum IV <lb/> +Invenire numero tria plana, quae bina juncta, ac etiam ipsa trium summa adscito dato plano, +quadratum constituant. +</quote> +<lb/> +<quote> +To find three plane numbers, of which the sum of any two, as also the sum of all three, +added to a given plane, constitutes a square. +</quote> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head233" xml:space="preserve" xml:lang="lat"> +Zet. 4: lib. 5. +<lb/>[<emph style="it">tr: +Zetetic 4, Book V. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1786" xml:space="preserve"> +Invenire numero tria plana quæ bina juncta ac etiam ipsa trium <lb/> +summa adscito dato plano Quadratum constituant. +<lb/>[<emph style="it">tr: +To find three plane numbers, of which the sum of any two, as also the sum of all three, +added to a given plane, constitutes a square. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1787" xml:space="preserve"> +Sit datum planum. +<lb/>[<emph style="it">tr: +Let the given plane be +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1788" xml:space="preserve"> +Aggregatum primi quæsiti <lb/> +plani et secundi +<lb/>[<emph style="it">tr: +The sum of the first plane sought and the second +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1789" xml:space="preserve"> +Aggregatum secundi <lb/> +et tertij +<lb/>[<emph style="it">tr: +The sum of the second and the third +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1790" xml:space="preserve"> +Summa trium +<lb/>[<emph style="it">tr: +The sum of the three +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1791" xml:space="preserve"> +Inde: summa <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo></mstyle></math> aggregato <lb/> +primi et secundi <lb/> +erit tertium planum +<lb/>[<emph style="it">tr: +Whence: the sum, minus the sum of the first and the second, will be the first plane. +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1792" xml:space="preserve"> +etiam: summa <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo></mstyle></math> aggregato <lb/> +secundi et tertij <lb/> +erit primum planum <lb/>[<emph style="it">tr: +also: the sum, minus the sum of the second and third, will be the first plane +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1793" xml:space="preserve"> +Inde: Aggregatum primi et <lb/> +tertij adscito plani <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mn>1</mn></mstyle></math> <lb/> +æquale <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>f</mi></mstyle></math> quadrato +<lb/>[<emph style="it">tr: +Whence: the sum of the first and third added to the plane <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mn>1</mn></mstyle></math> will equal the square <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>f</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f480v" o="480v" n="961"/> +<pb file="add_6782_f481" o="481" n="962"/> +<div xml:id="echoid-div324" type="page_commentary" level="2" n="324"> +<p> +<s xml:id="echoid-s1794" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1794" xml:space="preserve"> +This page is one of the few that contains a date: 29 August 1600. +The date is consistent with the suggestion that Harriot's friend Nathaniel Torporley met Viète in person +in Paris in the late 1590s, and brought Viète's books back to England (see Stedall 2003). <lb/> +This appears to be the first of several pages in which Harriot worked systematically through Viète's +<emph style="it">Zeteticorum libri quinque</emph> (1591 or 1593), +re-writing the propositions and proofs in his own notation. +He began here with Zetetics 2 and 3 from Book 1; Zetetic 1 is sketched in outline on Add MS 6782, f. 463. +He reached the end of Book I nine days later on 6 September (see Add MS 6782, f. 456.) +</s> +<lb/> +<quote xml:lang="lat"> +Zeteticum II <lb/> +Data differentia duorum laterum, & ratione eorumdem, invenire latera. +</quote> +<lb/> +<quote> +Given the difference of two roots, and their ratio, find the roots. +</quote> +<lb/> +<quote xml:lang="lat"> +Zeteticum III <lb/> +Data summa laterum, & ratione eorumdem: invenire latera. +</quote> +<lb/> +<quote> +Given the sum of two sides, and their ratio, find the sides. +</quote> +<lb/> +<s xml:id="echoid-s1795" xml:space="preserve"> +Viète used the letters <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> for the two roots, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>R</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>S</mi></mstyle></math> for their ratio. +Harriot repeated Viète's working in his own symbolic notation. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head234" xml:space="preserve" xml:lang="lat"> +Zetet. lib. 1. Zet. 2. 1600. August. 29. +<lb/>[<emph style="it">tr: +Zetetica, Book I, Zetetic 2. 1600, August 29. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1797" xml:space="preserve"> +Datur differentia duorum laterum et ratione eorundem invenire latera. +<lb/>[<emph style="it">tr: +Given the difference of two roots and their ratio, find the roots. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1798" xml:space="preserve"> +Primò. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> differentia <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> minus latus <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> maius latus <emph style="st">in</emph> ratione +latus minus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/> +ergo latus maius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>+</mo><mi>b</mi></mstyle></math>. <lb/> +Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>,</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>:</mo><mi>r</mi><mo>,</mo><mi>s</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +First. <lb/> +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> be the difference, <lb/> +and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> be the ratio of the smaller root to the larger root. <lb/> +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be the smaller root, therefore the larger root is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>+</mo><mi>b</mi></mstyle></math>. <lb/> +Therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>:</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mi>r</mi><mo>:</mo><mi>s</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1799" xml:space="preserve"> +Secundò <lb/> +latus maius. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. <lb/> +Ergo latus minus. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>-</mo><mi>b</mi></mstyle></math>. <lb/> +Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>,</mo><mi>e</mi><mo>-</mo><mi>b</mi><mo>:</mo><mi>s</mi><mo>,</mo><mi>r</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Second. <lb/> +Let the larger root be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, therefore the smaller root is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>-</mo><mi>b</mi></mstyle></math>. <lb/> +Therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>:</mo><mi>e</mi><mo>-</mo><mi>b</mi><mo>=</mo><mi>s</mi><mo>:</mo><mi>r</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<head xml:id="echoid-head235" xml:space="preserve" xml:lang="lat"> +Zet. 3. Data summa laterum, et ratione eorundem invenire latera. +<lb/>[<emph style="it">tr: +Zetetic 3. Given the sum of the roots and their ratio, find the roots. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1800" xml:space="preserve"> +primò. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. summa laterum <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>. minus latus <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>. maius latus <emph style="st">in</emph> ratione <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. esto maius latus +ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mo>-</mo><mi>a</mi></mstyle></math> latus minus. <lb/> +Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>,</mo><mi>g</mi><mo>-</mo><mi>a</mi><mo>:</mo><mi>r</mi><mo>,</mo><mi>s</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +First. <lb/> +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> be the sum of the roots, <lb/> +and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> be the ratio of the smaller root to the larger root. <lb/> +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be the larger root, therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mo>-</mo><mi>a</mi></mstyle></math> is the smaller root. <lb/> +Therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>:</mo><mi>g</mi><mo>-</mo><mi>a</mi><mo>=</mo><mi>r</mi><mo>:</mo><mi>s</mi></mstyle></math>. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1801" xml:space="preserve"> +secundò <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. esto minus latus. <lb/> +<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>-</mo><mi>g</mi></mstyle></math>. maius latus. <lb/> +Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>,</mo><mi>g</mi><mo>-</mo><mi>e</mi><mo>:</mo><mi>s</mi><mo>,</mo><mi>r</mi></mstyle></math>. +<lb/>[<emph style="it">tr: +Second. <lb/> +Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> be the smaller root, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>-</mo><mi>g</mi></mstyle></math> the larger root. <lb/> +Therefore: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>:</mo><mi>g</mi><mo>-</mo><mi>e</mi><mo>=</mo><mi>s</mi><mo>:</mo><mi>r</mi></mstyle></math>. +</emph>]<lb/> +[<emph style="it">Note: +Here <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>-</mo><mi>g</mi></mstyle></math> in the third line is clearly a writing or copying error for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mo>-</mo><mi>e</mi></mstyle></math>; +Harriot has proceeded correctly with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mo>-</mo><mi>e</mi></mstyle></math> in the fourth line. + </emph>]<lb/> +</s> +</p> +<pb file="add_6782_f481v" o="481v" n="963"/> +<pb file="add_6782_f482" o="482" n="964"/> +<pb file="add_6782_f482v" o="482v" n="965"/> +<pb file="add_6782_f483" o="483" n="966"/> +<pb file="add_6782_f483v" o="483v" n="967"/> +<pb file="add_6782_f484" o="484" n="968"/> +<pb file="add_6782_f484v" o="484v" n="969"/> +<pb file="add_6782_f485" o="485" n="970"/> +<pb file="add_6782_f485v" o="485v" n="971"/> +<pb file="add_6782_f486" o="486" n="972"/> +<pb file="add_6782_f486v" o="486v" n="973"/> +<pb file="add_6782_f487" o="487" n="974"/> +<pb file="add_6782_f487v" o="487v" n="975"/> +<pb file="add_6782_f488" o="488" n="976"/> +<pb file="add_6782_f488v" o="488v" n="977"/> +<pb file="add_6782_f489" o="489" n="978"/> +<pb file="add_6782_f489v" o="489v" n="979"/> +<pb file="add_6782_f490" o="490" n="980"/> +<div xml:id="echoid-div325" type="page_commentary" level="2" n="325"> +<p> +<s xml:id="echoid-s1802" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1802" xml:space="preserve"> +This page contains work on one of the ratios given by Viète under the heading 'Syntomon' +in Chapter XIX, Proposition 21 of <emph style="it">Variorum responsorum liber VIII</emph>. +For other pages based on the same (36, 45, 70)-degree spherical triangle see Add MS 6787, f. 36 to 39. +</s> +<lb/> +<s xml:id="echoid-s1803" xml:space="preserve"> +The reference to Regiomontanus is to his <emph style="it">De triangulis omnimodis</emph>, +Book 5, Proposition 2. For the same proposition, see also Add MS 6787, f. 51. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head236" xml:space="preserve" xml:lang="lat"> +Investigatio analogiæ <lb/> +Vietanæ +<lb/>[<emph style="it">tr: +An investigation of one of Viète's ratios +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1805" xml:space="preserve"> +(Investigavit Vieta <emph style="super">(ut putamus)</emph> per <lb/> +diagramma Regiomontani <lb/> +lib. 5. pr. de triangulis <lb/> +cum quibusdam additamentis, +ut nos <emph style="st">alibi</emph> etiam alibi.) +<lb/>[<emph style="it">tr: +Viète investigated this (as I believe) from a diagram of Regiomontanus in Book 5 of +<emph style="it">De triangulis</emph> with certain additions, as I have also done elsewhere. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1806" xml:space="preserve"> +ut Vieta <lb/> +pag: 47.b. +<lb/>[<emph style="it">tr: +As Viète, page 47v. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1807" xml:space="preserve"> +ut Vieta <lb/> +pag: 35.b. +<lb/>[<emph style="it">tr: +As Viète, page 35v. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1808" xml:space="preserve"> +conditiones <lb/> +alteri Δ<emph style="super">i</emph> <lb/> +contrariæ +<lb/>[<emph style="it">tr: +conditions in other, opposite triangles +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1809" xml:space="preserve"> +Nota. Et si signum (<) ponatur sub <emph style="st">ab</emph>, intelligetur quod ab < 90. <lb/> +Ita signum (>) sub <emph style="st">d</emph>, denotat d > 90. <lb/> +Istud signum, (< >), denotat unum latus maius alterum minus 90. &c. +<lb/>[<emph style="it">tr: +Note. If this sign (<) is placed under <emph style="st">ab</emph>, it is to be understood that ab < 90. <lb/> +This sign (>) under <emph style="st">d</emph>, indicates that d > 90. <lb/> +These signs, (< >), indicate that one side is greater than, the other less than 90. &c. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f490v" o="490v" n="981"/> +<pb file="add_6782_f491" o="491" n="982"/> +<div xml:id="echoid-div326" type="page_commentary" level="2" n="326"> +<p> +<s xml:id="echoid-s1810" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1810" xml:space="preserve"> +This page contains work on one of the ratios given by Viète under the heading 'Syntomon' +in Chapter XIX, Proposition 21 of <emph style="it">Variorum responsorum liber VIII</emph>. +For other pages based on the same (36, 45, 70)-degree spherical triangle see Add MS 6787, f. 36 to 39. +</s> +<lb/> +<s xml:id="echoid-s1811" xml:space="preserve"> +The reference to Regiomontanus is to his <emph style="it">De triangulis omnimodis</emph>, +Book 5, Proposition 2. For the same proposition, see also Add MS 6787, f. 51. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head237" xml:space="preserve" xml:lang="lat"> +Investigatio omnium <lb/> +analogiarum et casuum. +<lb/>[<emph style="it">tr: +An investigation of all the ratios and cases. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1813" xml:space="preserve"> +Æquationes conjugatæ <lb/> +sunt eadem: <lb/> +et una est <lb/> +impossibilis: <lb/> +sunt igiture <lb/> +tres tantum <lb/> +diversæ <lb/> +æquationes. <lb/> +et <lb/> +illæ, tres <lb/> +primæ. +<lb/>[<emph style="it">tr: +The conjugate equations are the same, and one is impossible; +there are therefore as many as three different equations, and of these, the first three. +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1814" xml:space="preserve"> +Ergo sunt tres analogiæ diversæ, et casus 6. +<lb/>[<emph style="it">tr: +Therefore there are three different ratios, and 6 cases. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f491v" o="491v" n="983"/> +<pb file="add_6782_f492" o="492" n="984"/> +<pb file="add_6782_f492v" o="492v" n="985"/> +<pb file="add_6782_f493" o="493" n="986"/> +<pb file="add_6782_f493v" o="493v" n="987"/> +<pb file="add_6782_f494" o="494" n="988"/> +<pb file="add_6782_f494v" o="494v" n="989"/> +<pb file="add_6782_f495" o="495" n="990"/> +<pb file="add_6782_f495v" o="495v" n="991"/> +<pb file="add_6782_f496" o="496" n="992"/> +<pb file="add_6782_f496v" o="496v" n="993"/> +<pb file="add_6782_f497" o="497" n="994"/> +<pb file="add_6782_f497v" o="497v" n="995"/> +<pb file="add_6782_f498" o="498" n="996"/> +<pb file="add_6782_f498v" o="498v" n="997"/> +<pb file="add_6782_f499" o="499" n="998"/> +<pb file="add_6782_f499v" o="499v" n="999"/> +<pb file="add_6782_f500" o="500" n="1000"/> +<pb file="add_6782_f500v" o="500v" n="1001"/> +<pb file="add_6782_f501" o="501" n="1002"/> +<div xml:id="echoid-div327" type="page_commentary" level="2" n="327"> +<p> +<s xml:id="echoid-s1815" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1815" xml:space="preserve"> +The reference at the top of this page is to Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book IV, Zetetic 4. +</s> +<lb/> +<quote xml:lang="lat"> +Zeteticum IV <lb/> +Invenire duo triangula rectangula simili datas habentes hypotenusas, +& diducti ab iis tertii trianguli basis, composita ex perpendiculo primi & base secundi, +erit ea quæ præfinitur. <lb/> +Oportebit autem basim illam præfinitam præstare hypotenusæ primi. +</quote> +<lb/> +<quote> +To find two similar right-angled triangles having given hypotenuses, +and subtracting the base of a third triangle, +composed of the perpendicular of the first and the base of the second, +will give a predefined quantity. +The predefined base, moreover, must be greater than the hypotenuse of the first triangle. +</quote> +<lb/> +<s xml:id="echoid-s1816" xml:space="preserve"> +Viète used the letters <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> for the hypotenuses of the first and second triangles, +and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi></mstyle></math> for the perpendicular of the third. +Harriot followed Viète's working but in his own lower-case notation. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head238" xml:space="preserve" xml:lang="lat"> +Zet. 4. 4. +<lb/>[<emph style="it">tr: +Zetetica, Book IV, Zetetic 4. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1818" xml:space="preserve"> +operatio fit <lb/> +per. 3am +<lb/>[<emph style="it">tr: +the operation may be done by the third +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f501v" o="501v" n="1003"/> +<pb file="add_6782_f502" o="502" n="1004"/> +<div xml:id="echoid-div328" type="page_commentary" level="2" n="328"> +<p> +<s xml:id="echoid-s1819" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1819" xml:space="preserve"> +The reference at the top of this page is to Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book IV, Zetetic 2. +This is also Proposition II.9 from the <emph style="it">Arithmetica</emph> of Diophantus, +but Harriot refers only to Viète's version of it. +</s> +<lb/> +<quote xml:lang="lat"> +Zeteticum II <lb/> +Invenire numero duo quadrata, aequalia duobus aliis datis quadratis. +</quote> +<lb/> +<quote> +To find in numbers two squares equal to two other given squares. +</quote> +<lb/> +<s xml:id="echoid-s1820" xml:space="preserve"> +One may interpret the problem as asking for two sets of Pythagorean triples with the same third side, +or two rational right-angled triangles with the same hypotenuse. +Viète called the sides of the first two squares <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>, and the hypotenuse <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. +The words 'synæreseos' and 'diæreseos' were used by Viète and are typical of the Greek terms +he frequently introduced into his writing. +Harriot followed Viète's working but in his own lower-case notation. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head239" xml:space="preserve" xml:lang="lat"> +Zet. lib. 4. 2. +<lb/>[<emph style="it">tr: +Zetetica, Book IV, Zetetic 2. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1822" xml:space="preserve"> +prop. 2. +<lb/>[<emph style="it">tr: +Proposition 2 +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1823" xml:space="preserve"> +via synæreseos +<lb/>[<emph style="it">tr: +by expansion +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1824" xml:space="preserve"> +via diæreseos +<lb/>[<emph style="it">tr: +by contraction +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1825" xml:space="preserve"> +prop. 1. +<lb/>[<emph style="it">tr: +Proposition 1 +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1826" xml:space="preserve"> +prop. 3. eadem ac secunda aliter +<lb/>[<emph style="it">tr: +Proposition 3. the same and the second another way. +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f502v" o="502v" n="1005"/> +<pb file="add_6782_f503" o="503" n="1006"/> +<div xml:id="echoid-div329" type="page_commentary" level="2" n="329"> +<p> +<s xml:id="echoid-s1827" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s1827" xml:space="preserve"> +The reference at the top of this page is to Viète's +<emph style="it">Zeteticorum libri quinque</emph>, Book IV, Proposition 3. +This is also Proposition II.9 from the <emph style="it">Arithemtica</emph> of Diophantus +(see also Add MS 6782, f. 502). +</s> +<lb/> +<quote xml:lang="lat"> +Zeteticum II <lb/> +Rursus, invenire numero duo quadrata, aequalia duobus aliis datis quadratis. +</quote> +<lb/> +<quote> +Again, to find in numbers two squares equal to two other given squares. +</quote> +<lb/> +<s xml:id="echoid-s1828" xml:space="preserve"> +One may interpret the problem as asking for two sets of Pythagorean triples with the same third side, +or two rational right-angled triangles with the same hypotenuse. +Viète called the sides of the first two squares <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>. +Harriot followed Viète's working but in his own lower-case notation. +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head240" xml:space="preserve" xml:lang="lat"> +Zet. lib. 4. 3. +<lb/>[<emph style="it">tr: +Zetetica, Book IV, Zetetic 3. +</emph>]<lb/> +</head> +<p xml:lang="lat"> +<s xml:id="echoid-s1830" xml:space="preserve"> +1. triang. +<lb/>[<emph style="it">tr: +triangle 1 +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1831" xml:space="preserve"> +2. Δ. simili. +<lb/>[<emph style="it">tr: +triangle 2, similar +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1832" xml:space="preserve"> +3. Δ. +<lb/>[<emph style="it">tr: +triangle 3 +</emph>]<lb/> +</s> +<lb/> +<s xml:id="echoid-s1833" xml:space="preserve"> +3. Δ. alterum. +<lb/>[<emph style="it">tr: +triangle 3, alternative +</emph>]<lb/> +</s> +</p> +<p xml:lang="lat"> +<s xml:id="echoid-s1834" xml:space="preserve"> +3. Δ. aliter. +<lb/>[<emph style="it">tr: +triangle 3 another way +</emph>]<lb/> +</s> +</p> +<pb file="add_6782_f503v" o="503v" n="1007"/> +<pb file="add_6782_f504" o="504" n="1008"/> +<pb file="add_6782_f504v" o="504v" n="1009"/> +<pb file="add_6782_f505" o="505" n="1010"/> +<p xml:lang="lat"> +<s xml:id="echoid-s1835" xml:space="preserve"> +Talia problemata hic +<emph style="super">schemata explicantur</emph> +<emph style="st">apponuntur</emph> quæ +<emph style="st">conducunt</emph> ad Magisteriorum <lb/> +formas conducunt intelligendas. <emph style="st">[???]</emph> similis. +</s> +</p> +<pb file="add_6782_f505v" o="505v" n="1011"/> +</div> +</text> +</echo> \ No newline at end of file