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The eclectic content and sources of Clavius’s Geometria Practica

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Abstract

We consider the Geometria Practica of Christopher Clavius, S.J., a surprisingly eclectic and comprehensive practical geometry text, whose first edition appeared in 1604. Our focus is on four particular sections from Books IV and VI where Clavius has either used his sources in an interesting way or where he has been uncharacteristically reticent about them. These include the treatments of Heron’s Formula, Archimedes’ Measurement of the Circle, four methods for constructing two mean proportionals between two lines, and finally an algorithm for computing nth roots of numbers.

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Notes

  1. See Knobloch (1988), Baldini (1983), Baldini (2003) for the original documentary sources for his life, career, and the activities of the Academy Clavius supervised at the Jesuit Collegio Romano. For a nuanced consideration of Clavius’s place in the development of the teaching of mathematics in the Jesuit schools, see Romano (1999).

  2. This is expressed most explicitly in Clavius’s essay In disciplinas mathematicas prolegomena (Prolegomena on the mathematical disciplines) included in Volume I of the Opera Mathematica, (Clavius 1611–1612). Clavius sees mathematics as intermediate between metaphysics and natural philosophy, an idea that traces back at least to Proclus’s Commentary on Book I of Euclid’s Elements, (Morrow 1970), a text Clavius mentions several times. See, for instance, Rommevaux (2005, Chapter 1).

  3. These have been digitized, see: https://clavius.library.nd.edu/mathematics/clavius.

  4. In the past, this was perhaps a reflection of certain derogatory attitudes toward “applied” or “practical” mathematics in general. Raynaud proposes in fact that practical geometry has been “... doublement marginal, vis-à-vis des mathematiques savantes et vis-à-vis des traditions techniques,” that is, “ ... doubly marginal, with respect to theoretical mathematics, and with respect to technical traditions.” (Raynaud 2015, p. 19)

  5. These features have also furnished the motivation for the author of this essay to undertake a translation of the entire Geometria Practica from the original Latin into English using the 1606 second edition. This translation is freely available at CrossWorks, the online faculty and student scholarship repository maintained by the Library of the College of the Holy Cross, at the URL: https://crossworks.holycross.edu/hc_books/57/. All quotations of passages from the Geometria Practica in English are taken from this translation. The original Latin text from the 1606 edition is provided in footnotes for purposes of comparison. The same will be done for quotations from other sources.

  6. The influence of Magini’s work is especially evident in Books II and III of Clavius (1606).

  7. Verum quoniam & hic de vnica tantum parte fuit sollicitus: & alii, quamuis aggressi omnia, multa tamen inter scribendum praeterierunt: decreui, si qua possem, perficere: vt, quicquid vtiliter in Geometria practica ab aliis traditum, à me etiam inuentum est, vnius operis gyro clauderetur.

  8. At a higher degree of granularity, the complete list of chapter headings and propositions that serves as the table of contents, see pages iv–xx of Clavius (1606), is even more evidence here.

  9. Discussions of problems similar to those considered here can be seen in almost all practical geometry books, although inspired by Magini’s treatment, Clavius’s collection of problems is much more extensive than many. As Raynaud points out, these are part of a long and surprisingly stable tradition with connections to propositions 19–22 from Euclid’s Optics, (Raynaud 2015, p. 15).

  10. Neque vero hoc praeter institutum nostrum existimare quis debet: cum per eiusmodi demonstrationes Geometricas studioso Lectori via multiplex aperiatur ad inuestigandas similes speculationes in rebus Geometricis: quippe cum in iis ad exercendum ingenium amplissimum campum habeat. (Clavius 1606, p. 330)

  11. As Knobloch writes, Clavius’s “... approche démontre les limites d’une division trop tranchée entre géométrie pratique et géométrie savante," (Knobloch 2015, p. 60). That is, Clavius’s “... approach shows the limitations of a too-definite division between practical geometry and theoretical geometry.” This applies to almost every section of the Geometria Practica, not just the discussion of geodesy.

  12. Et verò cum perpetua multorum annorum experientia compererim, admodum paucos esse, qui non in Mathematicis exerceantur eo consilio, vt quae didicerint, ad aliquem vsum trahant. (Clavius 1606, Preface)

  13. Etenim dum certa ratio traditur, qua camporum longitudines, altitudines montium, vallium depressiones, locorum omnium inaequalitates inter se, & interualla deprehendere metiendo debeamus: cuilibet liquet, vt arbitror, quantum commodi, vtilitatisque substructioni aedificiorum, cultui agrorum, armorum tractationi, contemplationi siderum, aliisque artibus, & disciplinis ex horum cogitatione manare possit.

  14. A more extensive version of this also appears in Clavius’s edition of Euclid.

  15. Clavius is often very careful to identify sources, and it stands out when he does not do so. Over the course of this book, the list of authors cited is quite extensive, including (but possibly not limited to) Apollonius, Archimedes, Archytas, Giovanni Battista Benedetti, Campanus de Novare, Girolamo Cardano, Federico Commandino, John Dee, Dinostratus, Diocles, Albrecht Dürer, Eratosthenes, Euclid, Eutocius of Ascalon, Oronce Fine, Fraçois de Foix, Comte de Candale, Niccolo Fontana (“Tartaglia”), Gemma Frisius, Marino Ghetaldi, Christoph Grienberger, Hippocrates, Hypsicles, Ioannes Pediasimos, Leonardo Pisano (“Fibonacci”), Ludolph van Ceulen, Mohammad of Baghdad, Odo van Maelcote, Giovanni Antonio Magini, Francesco Maurolico, Menaechmus, Nicholas of Cusa, Nicomedes, Latino Orsini, Luca Pacioli, Pappus, Georg Peuerbach, Proclus, Ptolemy, Joseph Justus Scaliger, Sporus, Simon Stevin, Theon of Alexandra, Juan Bautista Villalpando, Johannes Werner. A fuller listing of all the authors cited by Clavius across his whole written output is given in Knobloch (1990).

  16. But note that this would definitely be useful for measuring the area of a triangular plot of land or a triangular building where access to the interior and measurement of an altitude might not be possible.

  17. Hoc autem efficiam, si praescribam artem quandam generalem, qua cuiuscunque generis radicem extrahere possimus, ex libro eximij cuiusdam Arithmetici Germani depromptam fermè totam—(Clavius 1606, p. 276)

  18. Colligantur omnia latera in unam summam: Ex huius summa semisse subtrahantur singula latera, vt habeantur tres differentiae inter illam semissem, & latera singula: Postremo tres hae differentiae, & dicta semissis inter se mutuo multiplicentur. Producti enim numeri radix quadrata erit area trianguli quaesita.

  19. The Islamic mathematician al-Bīrūnī (973–1048) thought that the result was originally proved by Archimedes, and C. M. Taisbak has recently provided a conjectural reconstruction of the way Archimedes might have stated the result. See Taisbak (2014).

  20. This was first published in Schöne (1903). A modern study of this sole known surviving manuscript of the Metrica can be found in Acerbi and Vitrac (2014).

  21. See Fig. 1. To generate these figures, we used the triangle with vertices at \(B = (0,0)\), \(C = (5,0)\) and \(A = (1,2)\) in the Cartesian plane. This happens to have a right angle at A so some of the line segments in the figures are in rather special positions that facilitated the plotting. However, this does not affect the arguments. None of the authors we consider would have done things this way, of course.

  22. The difference in the diagrams is also noted by the translator B. Hughes in Pisano (2008). See the footnote on p. 83. But there is an unfortunate mistranslation at the start of the proof of Heron’s formula in Pisano (2008). At the start of the first full paragraph on p. 81, Hughes has, “To prove this: in triangle abg bisect the two equal angles abg and agb ... .” This would make the proof apply only to isosceles triangles. But that is not correct. The Latin text of Pisano (1862, p. 40) at this point is: “Ad cuius rei demonstrationem adiaceat trigonum abg: et dividantur in duo equa anguli, qui sub .abg. et .agb. a rectis .bt. et .tg. ... .” That is, “To prove this: in the triangle abg, let the angles abg and agb each be divided into two equal angles by the lines bt and tg ... . ” Fibonacci is definitely not restricting his discussion to isosceles triangles.

  23. For example, he was very explicit about this in the introduction to Book VI on geodesy (divisions of figures) where the material in question might ultimately derive from a lost work of Euclid, as discussed, for instance, in Knobloch (2015).

  24. The brevity of the work and its somewhat sketchy form have led Dijksterhuis to conjecture that “it is quite possible that the fragment we possess formed part of a larger work” (Dijksterhuis 1956, p. 222), and Knorr to judge that the versions we have represent “at best an extract from the original composition” (Knorr 1989, p. 375). Clagett (1964, 1967–1984, Volume 1) reproduces two translations of this work from Arabic into Latin, the first made (“perhaps”) by Plato of Tivoli (fl. twelfth century), and the second made by Gerard of Cremona. Clagett also reproduces six additional “emended” versions as well as the treatment of the results of this work in the Verba Filiorum, following the Banū Mūsā. Part III of Knorr (1989) contains a more complete study of the transmission including additional versions. By Clavius’s time, many versions of this work of Archimedes were available, including the Greek editio princeps published by Thomas Gechauff in 1544 and the Latin translation in Commandino’s edition of works of Archimedes (1558).

  25. That is, in modern terms, \(3\,\frac{10}{71}< \pi < 3\,\frac{1}{7}\). Archimedes may well have used methods similar to the ones to be discussed to produce tighter estimates for the ratio of the circumference to the diameter. But if so, no text doing this has survived.

  26. Non abs re ergo erit, si eius libellum de circuli dimensione acutissimum sane, & subtilissimum hic interferam, tum quia breuissimus est, quippe qui tribus duntaxat propositionibus constet: tum ne studiosus, vt rem tam vtilem, atque apud omnes artifices peruulgatam intelligat, Archimedem ipsum adire cogatur: tum vero maximè, quod cum Archimedis scripta ob affectam breuitatem, sint paulo obscuriora, illis nos lucem aliquam allaturos speramus.

  27. This proof has much in common with the proof of Proposition XII.2 of Euclid. It is also very closely related to the isoperimetric problems that Clavius will discuss later in Book VII.

  28. Haec autem detractio continua fiet, si primo loco auferatur ex circulo quadratum inscriptum ABCD. Hoc enim cum dimidium quadrati IKLM, circulo circumscripti, vt in schol. propos. 9. lib. 4. Eucl. ostendimus: circulus autem ipsius quadrati IKLM pars sit, erit quadratum inscriptum ABCD maius quam dimidium circuli. Deinde si auferantur à residuis quatuor segmentis quatuor triangula Isoscelia AOB, BPC, CQD, DNA, ductis rectis ad media puncta arcuum. Haec enim simul maiora sunt, quam dimidium quatuor segmentorum simul, cum vnum quodque maius sit, quam dimidium segmenti in quo existit. Completo enim rectangulo AR, [marginal note: 41. primi.] erit eius dimidium triangulum AND: ac proinde idem triangulum maius erit quam dimidium segmenti AND. Eademque ratio est de aliis. Pari ratione, si à residuis octo segmentis auferantur octo alia triangula Isoscelia in illis constituta,&c. atque ita deinceps.

  29. Clavius stops with the octagon so he has a diagram that has components virtually identical with the ones presented by Fibonacci and Pacioli. But many of Clavius’s point labels and additional lines constructed in a very “busy” diagram have been omitted at this stage for clarity.

  30. Scaliger, the eminent French Protestant classical philologist and historian, also fancied himself a mathematician. His mathematical magnum opus, grandly titled Cyclometrica Elementa, was published in 1594 in a lavish edition with statements of theorems in both Latin and ancient Greek. But more competent mathematicians quickly recognized that it was largely erroneous. In 1609, Clavius published an 84-page pamphlet Refutatio Cyclometriae Iosephi Scaligeri (Refutation of Joseph Scaliger’s claims about measurement of circles), giving a blow-by-blow analysis of all of the (numerous) errors in Scaliger’s work. This is contained as an appendix in Volume V of Clavius (1611–1612).

  31. Et sane miror, te, Mathematicus, cum sis, negare quantitatem aliquam illi esse aequalem, qua neque maior est, neque minor. Si enim aequalis non est, erit inaequalis. Igitur vel maior vel minor, contra hypothesim cum dicatur neque maior esse, neque minor. An non vides, non solum Archimedem, sed etiam Euclidem lib. 12. hunc argumentandi modum frequentissimè vsurpare?

  32. Clavius writes: Haec est Archimedis propositio 3. quam nos secundam facimus, vt doctrinae ordo servetur, quando quidem sequens propositio 3. quam ipse 2. facit, hanc nostram propositionem 2. in demonstrationem adhibet. (Clavius 1606, p. 185)

  33. The author used this expedient, in fact, in his translation of Clavius’s proof. In mathematical discussions, extreme verbosity is sometimes the unfortunate corollary of complete explicitness.

  34. Note that this essentially rescales the whole figure from the dimensions given before. Since it is the ratio of the lengths that is important, this is harmless.

  35. Posita igitur ce, 153. erit Ee, 306. Et si quadratum ipsius ce, 23409. dematur ex 93, 636. quadrato ipsius Ee, reliquum fiet quadratum ipisus Ec, 70, 227. cuius radix est paulo maior quam 265. ac proinde Ec, ad ce, maiorem habebit proportionem quam 265. ad 153. Secto iam angulo eEc, bifariam per rectam Ed, erit eE, ad Ec vt ed, ad dc. Et componendo, eE, Ec simul ad Ec, vt ec, ad dc. Et permutando, eE, Ec, simul ad ec, vt Ec, ad cd. Quia verò eE, Ec simul maiores sunt, quam 571. (quippe cum Ee sit 306. & Ec, paulo maior, quam 265.) & ec, posita est 153. habebunt eE, Ec, simul ad ec, maiorem proportionem quam 571. ad 153. ideoque & proportio Ec, ad cd, maior erit, quam 571. ad 153. ac proinde si cd, ponatur 153. erit Ec, paulo maior quam 571. Igitur quadratum ipsius Ec, paulo maius erit, quam 326, 041. atque idcirco, cum quadratum ipsius cd, sit 23, 409. erit quadratum ipsius Ed, quod quadratis rectarum Ec, cd, est aequale, paulo maius, quam 349, 450. eiusque radix maior quam \(591\, \frac{1}{8}.\) quippe cum huius radicis quadratum sit tantum \(349{,}428\, \frac{49}{64}.\) Habebit igitur Ed, ad, dc, maiorem proportionem quam \(591\, \frac{1}{8}.\) ad 153.

  36. See the figure inserted for the first time at Clavius (1606, p. 186) and repeated several times thereafter for the convenience of the reader. For comparisons, look at figures on folios 87 and 88 of Fine (1532) and the facsimiles in Knorr (1989, pp. 460, 463). Setting up the figure this way could have been a purely practical decision to reduce the number of different figures that had to be produced in printing the book.

  37. Circulus quilibet ad quadratum diametri proportionem habet, quam ad [sic] 11. ad 14. proximè. (Clavius 1606, p. 191)

  38. ... quae quidem area minus à vera distabit, quam illa, quae ex proportione Archimedis inuenitur. Sed quia difficilius est per magnos numeros calculum instituere, quam per minores, vsus artificum obtinuit, vt proportio Archimedis ad calculum adhibeatur. Quando tamen desideratur accuratior calculus, vtendum erit posteriori hac proportione Ludolphi, praesertim in maioribus circulis.

  39. About the proof of Proposition 3, Pacioli writes: “Ancora eglie da mostrare comme e fo trouata da Archimenide la linea circonferentiale essere .3. volte \(.\frac{1}{7}.\) del diametro: la quale inventione fo bella e sotile in questo modo, bene che con breuita se dica,” (Pacioli 1494), Pars Secunda, Distinctio quarta, Capitulum secundum, folio 31. Compare with Fibonacci’s introductory remarks given below. Clagett (1964, 1967–1984, Vol. 3, Part III) gives a closer comparison of Fibonacci’s version and Pacioli’s version.

  40. Clavius returns to this point in Book VIII of the Practical Geometry, discussing arguments by Archimedes from On the Sphere and Cylinder, and an alternate treatment by Girolamo Cardano.

  41. Quare concluditur, quod ex multiplicatione semidiametrij circulj in dimidium lineae circumferentis prouenit embadum ipsius. See Pisano (1862, p. 87).

  42. According to Clagett (1964, 1967–1984, Volume 3, Part III, p. 427), and referring to Pacioli’s version of this argument: “As given, this is very loose indeed.”

  43. fuit enim quadratum dyametrij suprascripti 196. et embadum ipsius 154. quorum proportio est sicut 14. ad 11. ... See (Pisano 1862, p. 88).

  44. Ostendendum est etiam quomodo inuentum fuit lineam circumferentem omnis circulij esse triplum et septima sui dyametrij ab Archimede philosopho, et fuit illa inuentio pulcra et subtilis ualde: quam etiam reiterabo non cum suis numeris, quibus ipse usus fuit demonstrare; cum possibile sit cum paruis numeris ea que ipse cum magnis ostendit plenissime demonstrare. See Pisano (1862, p. 88). Note that again there is no indication that 22/7 is only an approximation to the ratio in question.

  45. Inter datas duas rectas, duas medias proportionales prope verum inuenire

  46. For an extensive modern study of the surviving sources and the historical development, see Knorr (2012). Hippocrates of Chios (ca. 470–ca. 410 BCE) was traditionally credited with the reduction in the problem of duplicating the cube to the problem of constructing two mean proportionals between given lines. If AB and CD are the lines, two other lines XY and ZW are said to be two mean proportionals (in continued proportion) if

    $$\begin{aligned} AB {:} XY {:}{:} XY {:} ZW \quad \text {and} \quad XY {:} ZW {:}{:} ZW {:} CD. \end{aligned}$$

    Representing the lengths by numbers and using algebra, this becomes the string of equations

    $$\begin{aligned} \frac{AB}{XY} = \frac{XY}{ZW} = \frac{ZW}{CD}, \end{aligned}$$

    from which it follows that

    $$\begin{aligned} \left( \frac{ZW}{CD}\right) ^3 = \frac{AB}{CD}. \end{aligned}$$

    So for instance if \(CD = 1\) and \(AB = 2\) in some units, a construction of the two mean proportionals gives the line ZW which has length \(\root 3 \of {2}\), and that is the edge length of the cube with twice the volume of the cube with edge length \(CD = 1\).

  47. Quocirca prius in hac propos. in medium afferemus, quae antiqui Geometrae nobis hac de re scripta relinquerunt. Multorum enim ingenia res haec exercuit, atque torsit, quamuis nemo ad hanc vsque diem, verè, ac Geometricè duas medias proportionales inter duas rectas datas inuenerit.

  48. In the Latin: prope verum, literally “near the truth”.

  49. Constat ex his, qua ratione Cubus non solum duplicandus sit (quod veteres inquirebant) sed etiam augendus minuendusue in quacunque proportione: Item quo pacto pylae bombardarum maiores, aut minores fieri debeant secundum proportionem datam.

  50. German mathematician, 1468–1522.

  51. Praetermissis autem modis Eratosthenis; Platonis; Pappi Alexandrini; Spori; menechmi [sic] tum beneficio Hyperbolae, ac parabolae, tum ope duarum parabolarum; & Architae Tarentini, quamuis acutissimis, subtilissimisque; solum quatuor ab Herone, Apollonio Pergaeo, Philone Bysantio, Philoppono, Diocle, & Nicomede traditos explicabimus, quos commodiores, facilioresque, & errori minus obnoxios iudicauimus. Qui aliorum rationes desiderat, legere eas poterit in Commentarijs Eutocij Ascalonitae in librum 2. Archimedis de Sphaera, & Cylindro: Item in libello Ioannis Verneri Norimbergensis de sectionibus Conicis.

  52. ... hoc facili operari non possit, tamen, qualiter hoc fieri debeat. (Pisano 1862, p. 153).

  53. Protractis autem lateribus, DA, DC, intelligatur circa punctum B, moueri regula hinc inde, donec ita secet DA, DC productas in F, & G, vt rectae emissae EF, EG, aequales sint.

  54. The required circle is not shown in Fig. 8.

  55. Clavius does not use this name, though. In the coordinate system suggested by placing the diameters along the coordinate axes and taking the circle to have radius 1, the equation of the cissoid is \((x^2 + (y+1))^2 (y+1) = 2x^2\).

  56. Nicomedes construit prius instrumentum quoddam, quo lineam inflexam describit, quam Conchilem, vel Conchoideos appellat. Sed nos omisso eo instrumento, eandem, (quod ad nostrum institutum satis est) per puncta delineabimus, hac ratione (Clavius 1606, p. 270). Note the parallel with the discussion of the cissoid.

  57. More precisely, if we introduce coordinates placing the x-axis along the line AB and E at the origin and take \(CE = ED\), Nicomedes’ conchoid is one of the connected components of the real algebraic quartic curve defined by \((x^2+(y+1)^2)y^2 = (y+1)^2\). There is also a second connected component below the line AB with a cusp at the point \((0,-1)\), namely the point D. If some other ratio between the lengths CE and ED is specified, other conchoids with nodes at the point corresponding to D will be produced (Fig. 11).

  58. duas proprietates huius lineae insignes, (Clavius 1606, p. 270).

  59. In modern terms, the line AB is a horizontal asymptote of the conchoid.

  60. Dato quouis angulo rectilineo, & puncto extra lineas angulum datum comprehendentes: Ab illo puncto educere rectam secantem rectas datum continentes angulum, ita vt eius portio inter illas rectas intercepta aequalis sit datae rectae.

  61. The relevant passage from the original has been quoted in the Introduction.

  62. From its title, it might be expected that (Stifel 1545) is another candidate for Clavius’s source. But this is not the case. Even though this last book is written in German, unlike (Stifel 1544), its discussion of extracting roots (see folios 45 and 46) deals with abacus or counting board calculations, not the hand calculation algorithm that Clavius discusses.

  63. Clavius’s table is also pointed out in Knobloch (1988, p. 351), but Knobloch attributes this to the influence of Cardano and Tartaglia on Clavius. The identical and somewhat unusual formatting of the tables in both Clavius and Stifel is very suggestive.

  64. See Clavius (1606, p. 280). Sit ex numero

    $$\begin{aligned} \begin{array}{ccccccccc} 2 &{} \quad 3&{} \quad 9&{} \quad 4&{} \quad 8&{} \quad 3&{} \quad 1&{} \quad 9&{} \quad 0 \\ &{}&{}\bullet &{}&{}&{}\bullet &{}&{}&{}\bullet \end{array} \end{aligned}$$

    extrahenda radix cubica.

    Primvm ex puncto 239. subtraho cubum 216. qui est maximus in eo

    $$\begin{aligned} \begin{array}{rcr} 36&{}--&{}300\\ 6&{}--&{}30 \end{array} \end{aligned}$$

    contentus, cuius radicem 6. scribo in Quotiente ad marginem. Et quia relinquitur numerus . 23. erit sequens punctum 23483. Deinde paro diuisorem hoc modo. Supra radicem inuentam 6. pono eius quadratum 36. Et ad dextram colloco duos numeros peculiares radicis cubicae, nimirum 300. & 30. vt hic vides. Multiplico superiores duos numeros 36. & 300. inter se, & producto 10800. addo productum 180. ex multiplicatione numerorum inferiorum 6. & 30. inter se. Nam summa 10980. erit Diuisor. Satis etiam esset productus ex duobus superioribus inter se multiplicatis, nimirum 10800. pro Diuisore. quod in alijs extractionibus intelligendum quoque est. Diuido ergo punctum meum 23483. per diuisorem inuentum 10980. & Quotientum 2. scribo post figuram 6. prius inuentam.

    Pingo post haec figuram huiusmodi. Ad dextram numerorum 36. & 300.

    $$\begin{aligned} \begin{array}{rcrcr} 36&{}--&{}300&{}--&{}2.\\ 6&{}--&{}30&{}--&{}4.\\ &{}&{}&{}&{} 8. \end{array} \end{aligned}$$

    colloco inuentam figuram 2. & infra eam eius quadratum 4. & sub hoc cubum eiusdem 8. Nam si tam superiores tres numeri 36. 300. & 2. quam inferiores tres 6. 30. & 4. inter se multiplicentur, & productis 21600. & 720. addatur cubus 8. fiet numerus 22328. quem si ex meo puncto 23483. subtraham, remanent 1155. atque adeo puncto sequens erit 1155190.

  65. We have chosen to translate this passage from the German of Rudolff and Stifel (1553) rather than one of the parallel computations in Stifel (1544), which are written in Latin. Clavius insists on the German provenance of the method, so he may well have learned it first from here, although from various features of his later Algebra text (1608), it is clear that he was also familiar with Stifel (1544).

    Exemplum.

    $$\begin{aligned} \begin{array}{cccccccccccccccccccc} &{}\bullet &{}&{}&{}\bullet &{}&{}&{}\bullet &{}&{}&{}\bullet \\ 8 &{} 0 &{} 6 &{} 2 &{} 1 &{} 5 &{} 6 &{} 8 &{} 0 &{} 0 &{} 0 \end{array} \end{aligned}$$

    Erstlich subtrahir ich von dem hindersten puncten (das ist von 80) die aller gröste cubic zal/ die ich subtrahiren kan. Die selbig ist 64. so bleybett nach vbrig davon 16 die gehören denn sum nehisten puncten hernach/ der selbig uverkompt denn dise figuren 16621. So setz nu die cubic würzel von 64 in den quotient. facit 4. und is also der erst punct aufsgericht.

    So nehme ich nu fur mich den andern punct/ nemlich 16221. Den dividir ich mit 4800. (das kompt von 300 mal 16) Nu gibt das gedacht dividiren nur 3 in den quotient. Und ist also die newe figur gefunden.

    Dem selbigen nach stehn die zwo zalen 300 und 30. mit jren zugethonen zalen also.

    $$\begin{aligned} \begin{array}{rcrcr} 16&{}--&{}300&{}--&{}3\\ 4&{}--&{}30&{}--&{} 9\\ \end{array} \end{aligned}$$

    Denn erstlich ist gefunden in den quotient de figur 4. die steht neben 30 zur lincken hand/ vnd drob neben 300 steht jr quadrat/ nemlich 16.

    So is nu darnach gfunden in den quotient die figur 3. Die steht oben neben 300 zur rechten hand/ vnd darunder steht jr quadrat 9. neben 30. wie du alles vol sihest.

    So multiplicir ich nu/ vnd sprich. 16 mal 300 mal 3. facit. 14, 400. vnd 4 mal 30 mal 9. facit 1080. Das addir ich/ so kompt 15, 480. Das subtrahir ich von 16, 621. Als vom andern puncten diser operation/ so bleyben denn 1141.

    Auffs letzt multiplicir ich die newe gefundne figur Cubice. Nemlich 3 mal 3 mal 3. facit 27. die subtrahir ich auch/so bleyben 1141. die gehören zu volgenden punct.

  66. It is true, though, that Stifel’s works appeared several times in the Indices of works to be forbidden or expurgated prepared by Catholic authorities in various locations in the mid-to-late 16th and early 17th centuries [including Venice in 1554, Antwerp in 1571, (Gijón 2015, p. 80), and Spain in 1632 (and possibly earlier)]. What this entailed in the case of the 1632 Expurgatorio of the Spanish Inquisition is visible in the digitized copy of Stifel (1544) from the library of the Universidad Complutense de Madrid at https://babel.hathitrust.org/cgi/pt?id=ucm.5323774127. This book shows extensive hand-written emendations and striking-out of honorifics, etc. In particular, the hand-written notation “autore damnato, opus permissum” appears next to Stifel’s name on the title page. While Stifel himself had been anathematized, it was still permissible to read the mathematical contents of the work.

  67. In the Ordo Servandus in Addiscendis Disciplinis Mathematicis, essentially a position paper on the role of mathematics instruction prepared during the development of the Ratio Studiorum, see Gatto (2006, p. 252).

  68. In Kepler’s Harmonice Mundi. The reference was to Clavius’s discussion of various approximate constructions of regular heptagons in Book VIII.

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Little, J.B. The eclectic content and sources of Clavius’s Geometria Practica. Arch. Hist. Exact Sci. 76, 391–424 (2022). https://doi.org/10.1007/s00407-022-00288-5

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