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Paris, BnF, lat. 16657 · 84r

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9 9] 4 K secundorum 55 55] corr. ex ... sollers practicus inveniet, est enim ea corda latus decagoni; cordam vero pentagonicam que arcui 72 72] corr. ex ... graduum subtenditur componi ex partibus 70 punctis 32 32] corr. ex ... et secundis fere tribus; sed et latus exagoni supra quod arcus 60 graduum curvatur 60 itidem partibus terminari. Ad eundem quoque modum quia latus tetragoni, existens 90 partium corda, quadratum medie dyametros dyametros] While this looks accusative, it actually is functioning as a singular genetive. duplat potentialiter, latus item trigonale ens 120 120] corr. ex ... graduum corda medie dyametros quadratum potentialiter triplat, illud quidem partibus 84 punctis 15 15] 51 K secundis 10 fere concludi, illud autem partibus 100 et tribus punctis 55 55] corr. ex ... secundis 23 23] corr. in 33 (perhaps other hand). The ‘23’ is correct mathematically. equari, diligens examinator compariet compariet] This appears to be an unorthodox form of the verb ‘comparo.’ manente dico dico] This actually says ‘dice’ but any reader would realize the correct reading. predicta dyametri in 120 120] corr. ex ... equas porciones porciones] corr. ex ditones (perhaps other hand) sectione. Adhuc Adhuc] perhaps ad hec ex eodem teorumate cum sit corda arcui 36 subtensa ex partibus 37 punctis 4 secundis 55, 55] corr. ex ... cordam que residuo arcui de semycirculo, scilicet arcui 144 graduum, ⟨subtenditur⟩ subtenditur] From K partibus 124 124] 114 N graduum…124] i. m. (perhaps other hand) punctis 7 secundis 37 fere terminandam esse sobrius vestigator agnoscet.

Amplius exequentium demonstratione constat ad ad] corr. in ex certorum arcuum differentias differentias] corr. in differentiis cordas multas posse inveniri. Qua quidem ratione corda arcus 12 12] The original Arabic numerals, which are in an unusual form, are deleted, and the number is written in more typical Arabic numerals above the line. graduum reperienda est, hiis inquam qui qui] que K sunt arcuum 60 atque 72 72] The original Arabic numerals, which are in an unusual form, are deleted, and the number is written in more typical Arabic numerals in the margin. cordis precognitis. Deinceps plurimas diversorum arcuum cordas invenire inventas secundum arcum mediare sciemus, utpote arcus 12 12] The original Arabic numerals, which are in a more unusual form, are deleted, and the number is written in more typical Arabic numerals above the line. partium cordam, et deinde arcus 6 partium, nec minus quoque trium, eius tunc qui habet partem et dimidiam, et deinde qui ex media et quarta constabit.

Docet autem hec observatio unius partis et medie cordam ex parte una punctis 34 secundis 15 15] The 5 appears to have been corrected from a more unusual form to a more usual one. constare, retenta dico dicta dyametri divisione; ad eundem denique modum arcus medie partis et quarte corda corda] cordam K puncta 47 habere secunda 8. Amplius. Ex sequenti apodixi ratum est secundum arcum unius partis et medie et eius cordam quamlibet cordam multiplicis arcus posse inveniri. Nam eo arcu duplicato vel triplicato et deinceps omnes corde note occurrunt.

Verum cordam unius gradus sub certa veritate nulla deprehendit ratio. Quamvis enim ad arcum unius gradus et medii corda constiterit, eius tercie partis corda sub numeri compoto nullatenus scibilis est. Eius tamen rei notitia presenti intentioni necessaria est. Summo igitur studio et industria, quamvis non verissime tamen omnis sensibilis erroris periculo depulso, unius gradus corda per cordam unius gradus et medii sed etiam per medii et quarte in hunc modum reperta est.

Protrahimus in circulo cordam AB unius partis, AG vero unius gradus et medii. Quemadmodum ergo supradictum est quia AG ad AB quam arcus maioris ad arcum minoris minor est proportio. AG autem arcus ad AB arcum sexquialter est, linea ergo AG ad AB necessario quam sexquialtera minor erit. Constat autem cordam AG gradum unum puncta 34 secunda 15 15] The 5 appears to have been rewritten over the original. habere; unde corda AB maior quam gradus et puncta 2 2] The original Arabic numeral, which is in an unusual form, is deleted, and a more typical Arabic numeral is written above the line. secunda 50 50] The 5 appears to have been corrected to a more typical form over the original. profecto constabit. Unus namque gradus cum 34 punctis et secundis 15 gradum unum puncta 2 2] A more unusual form is deleted and replaced with a more usual numeral form. secunda 50 50] The 5 was corrected to a more typical form over the original. integraliter sexquialterat. Rursus AB lineam arcus medii gradus et quarte, ipsam vero vero] sup. lin. (perhaps other hand) AG ad unum gradum cordam statuimus. Igitur arcus AG ad AB sequitercius sequitercius] sesquitertius K est. Sed palam ex supradictis cordam AB punctis 47 secundis 8 concludi.

Sed ad hunc numerum scilicet puncta 47 secunda 8 sexquitercius numerus est hic, pars una puncta 2 2] The unusual form is deleted, and a more usual form is written over the line. secunda 50 tercia 40. Ergo corda unius gradus maior est quam pars una puncta 2 secunda 50 et minor quam pars una‌1…pars‌2] i. m. (perhaps other hand) una puncta 2 secunda 50 tercia 40. Non est ergo incongruum cordam unius gradus ponere partem unam puncta 2 secunda 50, quia minus quam in duabus terciis unius tercii tercii] This should read ‘secundi.’ The error is known to be less than 40’’’, i.e. 2/3 of a second. error erit, quare multo minus quam in uno secundo, sed ⟨in⟩ in‌2] From B inquisitione cordarum quod minus quam secundum fuerit postponitur. Sed…postponitur] One class of manuscripts has an alternate text for this paragraph.

Unde manifestum quoniam arcus dimidii gradus corda punctis 31 secundis 15 15] To follow Ptolemy’s table of chords, this number should be ’25.’ fere concluditur. Ad cuius quantitatis exemplar reliquas que inter duas certas cordas binatim cadunt possumus sine sensibili errore deprehendere. Namque duorum graduum cordam eius que est dimidii ad unius et dimidii facit cognosci adiectio.