“Familiar Demonstrations in Geometry”: French and Italian Engineers and Euclid in the Sixteenth Century

On the military revolution debate, see Parker Geoffrey , “The military revolution, 1560–1660, a myth?”, Journal of modern history, xlviii (1976), 195–214, and The military revolution: Military innovation and the rise of the West, 1500–1800 (Cambridge, 1988). See also Eltis David , The military revolution in sixteenth-century Europe (London, 1995), and above all Rogers Clifford , The military revolution debate: Readings on the military transformation of early modern Europe (Boulder, 1995).

On the influence of the Italian ways of fighting upon European warfare, see Mallett M. E. , Mercenaries and their masters: Warfare in Renaissance Italy (London, 1974), especially chapters 7 and 9. See also Hale John R. , War and society in Renaissance Europe, 1450–1620 (London, 1985). On the Turkish threat, see Luigi Zangheri, “Gli architetti italiani e la difesa dei territori dell'Impero minacciati dai turchi”, in Atti del convegno di studi architettura militare nell'Europa del XVI secolo, ed. by Cresti Carlo , Amelia Fara and Daniela Lamberini (Siena, 1988), 243–53.

On Venice and war, see Chambers David S. Clough Cecil H. Mallett Michael E. , (eds), War, culture and society in Renaissance Venice: Essays in honour of John Hale (London, 1993), and Mallett M. E. Hale John , The military organization of a Renaissance state: Venice c. 1400 to 1617 (Cambridge, 1984). On the concept of territorial machine, see Concina E. , La macchina territoriale: La progettazione della difesa nel cinquecento Veneto (Rome and Bari, 1983).

See the chapter entitled “Military authorship in a century of conflict” in Long Pamela , Openness, secrecy, authorship: Technical arts and the culture of knowledge from Antiquity to the Renaissance (Baltimore, 2001).

Biagioli Mario , “The social status of Italian mathematicians, 1450–1600”, History of science, xvii (1989), 41–95.

Johnston's Stephen “Making mathematical practice: Gentlemen, practitioners and artisans in Elizabethan England”, Ph.D. thesis, Cambridge, 1994, is available online at http://www.mhs.ox.ac.uk/staff/saj/thesis/abstract.htm. See also from the same: “The identity of the mathematical practitioner in 16th-century England”, in Hantsche Irmgarde (ed.), Der “Mathematicus”: Zur Entwicklung und Bedeutung einer neuen Berufsgruppe in der Zeit Gerhard Mercators (Duisburger Mercator-Studien, iv; Bochum , 1996), 93–120, as well as “Like father, like son? Dee John Digges Thomas and the identity of the mathematician”, in Clucas Stephen (ed.), John Dee: Interdisciplinary studies in English Renaissance thought (International Archives of the History of Ideas / Archives Internationales d'Histoire des Idées, 193; Dordrecht, 2006).

Walton Steven A. , “The mathematical and military sciences in Renaissance England”, Endeavour, xxiv (2000), 152–6.

Lanteri Giacomo , Due dialoghi di M. Iacomo de Lanteri da Paratico, Bresciano, ne i quali s'introduce messer Girolamo Catanio Novarese, & messer Francesco Trevisi ingegnero Veronese, con un giovene Bresciano, a ragionare del modo di disegnare le piante delle fortezze secondo Euclide; et del mode di comporre i modelli & torre in disegno le piante delle citta (Venice, 1557). The context of this publication is developed by Long, op. cit. (ref. 4), 203–5.

One particular source to identify these people is the Dictionary of Italian biographies, the other being the various works of Carlo Promis, an Italian scholar from the nineteenth century who wrote a number of books on Italian engineers.

For biographical information concerning Iacomo Lanteri, see Promis Carlo , “Della vita e delle opere degli scrittori di artiglieria, architetura e meccanica militare da Egidio Colonna a Francesco Marchi”, in Francesco di Giorgio Martini, Trattato di archittetura civile e militare (Turin, 1841), 95–98 and 110–12.

The surviving texts written by Giacomo Lanteri are, besides his already-mentioned Due dialoghi, his Due libri di M. Giacomo Lanteri di Paratico da Brescia. Del modo di fare le fortificationi di terra intorno alle citta & alle castella per fortificarle, et di fare cosi i forti in campagna per gli allogiamenti degli esserciti (Bologna, 1559), and his Due libri … del modo di fare le fortificationi di terra intorno alle città, & alle castella per fortificarle… (Venice, 1559). Another book published later in Venice, written in collaboration with Giovanni Battista de' Zanchi, Antonio Lupicini, Tommaso Baglioni, and Roberto Meietti, is entitled Della fortificationi. Delle offese et diffese delle città et fortezze di Giacomo Lanteri Bresciano, Gieronomi Zancho da Pesaro. Con due discorsi d'architettura militare d'Antonio Lupicini fiorentino (Venice, 1582).

On Girolamo Cataneo's life, see the article by L. Oliviato in the Dizionario biographico Italiano, xxii (1979), 471–3.

See Lanteri Iacomo , Due dialoghi di M Iacomo Lanteri da Paratico, Bresciano ne i quali s'introduce messer Girolamo Catanio, Novarese, & messer Francesco Trevisi, ingegnere Veronese, con un giovene Bresciano, a ragionare del modo di disegnare le piante delle fortezze secondo Euclide; et del modo di comporre i modelli & torre in disegno le piante delle citta (Venice, 1557). The figures appearing in the dialogues are Marcantonio Moro, Hercole Rozzone, Giovan Battista Rozzone, Marcantonio Calino, Gabriello Gandino, Vincenzo Gabiano and Marcantonio d'Acqua Viva, Conte Camillo d'Arco, Conte Aloisio Avogadro, Francesco Maria della Rovere, Duca di Urbino, il Duca di Ferrare, Girolamo and Curtio Martinengo, Gianbattista Gavardo, Pompilio Luzago, Nicolo Seccho (ambassador of Ferdinand to the Great Turk), and, last but most important for Lanteri, the Earls Francesco (†1571) Felix and Oliviero d'Arco. This circle may be completed with other names quoted by Cataneo in an other text, his Libro nuovo di fortificare (Brescia, 1567): The Earls Girolamo and Albrigo di Lodrone, the Earl Battista Vinciguerra, the Colonel Vincenzo Tadei, the Earl of Porcia and Brugnara, and again the Earls Giovann Battista and Curtio Marinengo. Cataneo was later published in French under the title: Le capitaine de Jérosme Cataneo, contenant la maniere de fortifier places, assaillir et defendre…. Le tout reveu, corrigé et augmenté en plusieurs lieux par l'auteur, et depuis mis en François [by Jean de Tournes] (Lyon, 1574).

See Tartaglia Niccolò , Quesiti et inventioni diverse de Nicolo Tartaglia. Di novo restampati con una aggionta al sesto libro, nella quale si mostra duoi modi di redur una città inespugnabile (privately published, 1554). For a French translation of Tartaglia's Nova scientia and of the Quesiti, see La balistique de Nicolas Tartaglia, first published in 1537 under the title “La science nouvelle”, and continued in 1546 in the first two books of Recherches et inventions diverses relatives à l'artillerie par Nicolas Tartaglia, translated from the Italian by Rieffel (Paris, 1845–46). There is also a partial English translation in Drake Stillman Drabkin I. E. (eds), Mechanics in sixteenth-century Italy: Selections from Tartaglia, Benedetti, Guido Ubaldo, and Galileo (Madison, 1969).

“Guido Taddini, Vita di Gabriele Tadino di Martinengo, Priore di Barletta”, Atti del Ateneo di Scienze Lettere et Arti, suppl. to vol. xxxvi (1975), and Giovan Battista Gallizioli, Memorie per servire alla storia della vita di Gabriele Tadino, Priore di Barletta (Bergamo, 1783).

Cataneo , Libro nuovo di fortificare (ref. 13), 2 (“Avvertimenti intorno alla presenta opera”), my translation.

There are two texts written by Ambroise Bachot: Le timon du Capitaine A-B. lequel conduira le lecteur parmi les guerrières mathématiques sur la réduction des unes aux autres figures géométriques et instruments de mesurer toutes distances et représenter en perspective joint un traicté fort utile des fortifications machines de guerre et autres particularités inventées par l'auteur (Paris, 1587), and Le gouvernail d'Ambroise Bachot capitaine ingénieur du Roy lequel conduira le curieux de géométrie en perspective dedans l'architecture des fortifications, machines de guerre et plusieurs autres particularités y contenues (Melun, 1598). On Bachot, see Gnudi M. Teach , “Agostino Ramelli and Ambroise Bachot”, Technology and culture, xv (1974), 1974–25, and Hélène Vérin, La gloire des ingénieurs (Paris, 1993), the most extensive study of Bachot's work.

He was also related to Jean Errard, another expert on fortifications. Fredéric Métin, who teaches mathematics in Dijon, is currently working on Errard's Geometry.

Biographical elements on Castriotto can be found in his Della fortificatione delle Citta di M. Girolamo Maggi e del capitan Iacomo Castriotto (Venice, 1564). See also Rocchi E. , Le fonti storiche dell'architettura militare, and Promis C. , Dell'arte dell'ingegnere e dell'artiglieria in Italia dalla sua origine fino al principio del XVI secolo (Memorie storiche; Turin, 1841).

Lanteri , Due dialoghi (ref. 13), 4, my translation.

Colliado Luigi , Pratica manuale della artiglieria dove si tratta dell'eccelenza & origine dell'arte militare e delle machine usate dagli antichi, composta da Luigi Colliado Ingegnero del Real'Essercito di S Maesta Catolica in Italia (Milan, 1606).

Asch Eric , in Power, knowledge and expertise in Elizabethan England (Baltimore, 2004), has developed the idea that mathematicians in Elizabethan England invented the concept of expert mediator and that this was linked to the increasing centralization of power. It is clear that this was a general evolution that started long before Elizabeth and concerned all Europe.

See for instance in England, Barret Robert , The theorike and practike of moderne warres (London, 1598) or, for the Spanish side, Don Sancho de Londoño, who, in his Discuros sobre la forma de reducir la disciplina militar a mejor y antiguo estado (Brussels, 1587), like Francisco de Valdès in his Espejo y disciplina militar (Brussels, 1586), defends the need for theoretical learning in the military profession.

See Bachot , Le gouvernail (ref. 17), 1, my translation.

I am here grateful to Daniela Lamberini for pointing out to me that the Galussi mentioned by Horst de la Croix was no other than Giovann Battista Belluzzi. See de La Croix Horst , “The literature on fortification in Renaissance Italy”, Technology and culture, iv (1963), 30–50, and Lamberini Daniela , “Il trattato delle fortificazioni di terra [Belluzzi (G.B.)]”, in Il disegno interrotto trattati medicei d'architettura, ed. by Borsi F. (Florence, 1980), i, 375–531; ii, 135–55.

Shapin Steven , “A scholar and a gentleman: The problematic identity of the scientific practitioner in early modern England”, History of science, xxix (1991), 279–327. For Guidobaldo dal Monte, see Henninger-Voss Mary , “Working machines and noble mechanics: Guidobaldo del Monte and the translation of knowledge”, Isis, xci (2000), 2000–59.

Bachot , Le gouvernail (ref. 17), 1, my translation.

Bachot , Le gouvernail (ref. 17), 3.

Lanteri , Dialoghi (ref. 8), 6, my translation.

Ibid., 8, my translation.

Tartaglia , Quesiti (ref. 14), 7, my translation.

Ibid., 7.

Ibid., 8.

Ibid., 12.

On Tartaglia's life, see Charbonnier Pierre , Essais sur l'histoire de la ballistique (Paris, 1928), 8–37. On Della Rovere's fascination for firearms, see for instance in Pesaro in the Bibliotecca Oliveriana, Ms 969, Disegni di artiglierie dei Della Rovere.

Lanteri , Dialoghi (ref. 8), 52.

Bachot , Le gouvernail (ref. 17), my translation. The source for this quotation may be found in the chapter entitled Discours des mesurables, f. 1 (not paginated).

Ibid.

Cuomo Serafina , “Shooting by the book: Notes on Niccolò Tartaglia's Nova Scientia”, History of science, xxxv (1997), 155–88. See also Rose P. L. , The Italian renaissance of mathematics (Geneva, 1975), and especially the chapter entitled: “The new science of Tartaglia and Benedetti”, 151–8.

Cataneo Girolamo , Tavole brevissime per sapere con prestezza quante file vanno a formare una giustissima battaglia. Con li suoi armati di corsaletti da cento fin a ventimilia huomini. Et appresso un facilissimo, et approvato modo di armarla di archibugieri, & di ale di cavalleria secondo l'uso moderno. Di nuovo aggionte, et largamente ampliate, tanto nella dichiaratione, come in esse tavole, dal medesimo auttore (Brescia, 1567).

See the Quesiti, Book 4, and for Da Coniano's tables see Maggi Girolamo , Della fortificatione delle città, di Girolamo Maggi, e del capitan Jacomo Castriotto…. Discorso del medesimo Maggi sopra la fortificatione de gli alloggiamenti de gli esserciti. Discorso del capitan Francesco Montemellino sopra la fortificatione del Borgo di Roma. Trattato dell'ordinanze, ò vero battaglie del capitan Giovacchino da Coniano. Ragionamento del sudetto Castriotto sopra le fortezze della Francia (Venice, 1583).

Among the authors publishing such paper instruments, see for instance in England Robert Barret, The theorike and practike of moderne warres (London, 1598), or for the Spanish side Francisco de Valdès, Espepo y disciplina militar (Brussels, 1586).

Sixteenth-century officers used special techniques to extract square roots, based on the ancient system of the gnomon or on a complex operation involving “offset double products”, a method based on the idea that the two digits of the root of a number Δ would correspond to the unknown quantities m and n satisfying the equality Δ = (10m + n)². For an explanation of this method, see Digges Thomas , Stratioticos (London, 1579), 14 et seq., or a manuscript to be found in the Bibliotecca Nazionale, Firenze, Ms Magliabecchi, Classe xix, Codice 13: Papinio Leopardo's compendio militare, 1570 (?).

Lanteri , Dialoghi (ref. 8), 89.

See Le quattro parti de le novelle del Bandello, ed. by Balsamo-Crivelli Gustavo , ii (Turin, 1910), 17: “Messer Niccolo quel di ci tenne al sole più di due ore a bada, per ordinar tre mila fanti secondo quell'ordine che aveva scritto, e mai non gli venne fatto di potergli ordinare.” Bandello's denunciation of Machiavelli's amateurism should be however considered with caution, since it is an ancient topos to oppose the intellectually minded scholar to the efficient practitioner. Nervertheless, Bandello's prejudice against Machiavelli is shared by Pierre Bourdeille, seigneur de Brantome, in his Grands capitaines françois, couronels françois, oeuvres complètes, v (Paris, 1879).

On “quadrat battel of ground”, see Tartaglia , Quesiti et inventioni diverse (ref.14), 43–44.

See for instance Styward Thomas , The pathwaie to martial discipline (London, 1582), 95.

See for instance the description of Cerisolles's battle by Blaise de Montluc, in his Commentaires, 1521–1576 (Paris, 1964), 156–67.

Barret Robert , for instance, in his The theorike and practike of moderne warres (London, 1598), 47, writes about semi-circular formations (half moons) of musketeers he has seen employed by Spaniards in the Low Countries, and Digges in his Stratioticos also alludes to his battlefield experience.

Cox Virginia , The Renaissance dialogue: Literary dialogue in its social and political contexts, from Castiglione to Galileo (Cambridge, 1992), writes (p. 34): “Lanteri's use of the dialogue can only be ascribed to his desire to show that, in an age in which meccanico was a voce d'ingiuria, the theory at least of his mechanical art was a subject worthy of highly civilized men. Certainly this is the only possible explanation apart from that of blatant ingratiation, which can account for the narrated dialogue with which Cataneo opens his discourse on geometry in the first dialogue. His account of a conversation on the subject between himself and his patrons is, as argument, spectacularly otiose, being mainly occupied with expressions of his riverenze and the cortesie of his listeners. But its subliminal message is clear enough: That the theory of military architecture is a subject of the greatest interest to the nobilissimi Conti d'Arco, and that, therefore, if it is frequently insulted and despised, as a mechanical art, that is only because of the false snobbery of the ignorante vulgo.”.

Lanteri , Dialoghi (ref. 8), 8, my translation.

Ibid., 10.

Ibid., 5. “Signori risposi io egli fa di mestieri (a colui che di cio vole perfettamente esser instrutto) sapere le proportione i sei primi libri d'Euclide; perche per via di quelli si puo d'ogni maniera di piante benissimo trattare”.

Bachot , Le gouvernail (ref. 17), 1, my translation.

Lanteri , Dialoghi (ref. 8), 51.

Ibid., 10. “Perche di sopra vi senti ricordare il quarto d'Euclide, m'è venuro in mente (se bene mi si ramenta) che il detto quarto benissimo insegna à descrivere le figuri di lati & d'angoli eguali”.

Bachot , Le gouvernail (ref. 17), 4.

Shelby Lon R. , “The geometrical knowledge of medieval master masons”, Speculum, xlvii (1972), 295–421. For the texts themselves, see de Honnecourt Villard , Carnet, XIIIe siècle, with comments by Alain Erlande-Brandenburg, Régine Pernoud, Jean Gimpel and Roland Bechman (Paris, 1986), and Recht Roland , “Le traité de géométrie de Mathieu Roriczer”, Histoire et archéologie, Dossier 47 (1980), 24–25. Jeanne Peiffer remarks on the influence of those treatises on Dürer in her Albrecht Dürer, géométrie (Paris, 1995).

Bachot , Le gouvernail (ref. 17), 6.

Bachot , Le gouvernail (ref. 17), 8.

Bachot , Le gouvernail (ref. 17), 2.

Lanteri , Dialoghi (ref. 8), 60.

Lanteri , Dialoghi (ref. 8), 6.

Cf. Reysch Gregor , Margaritha philosophica (Freiburg, 1503).

Opus elementorum Euclidis in geometriam artem. In id quoque Campani commentationes. Preclarissimus liber elementorum Euclidis perspicacissimi: In artem geometriæ incipit … (Venice, 1482). A short history of the editions of Euclid in the Renaissance may be found in Bernard Vitrac's introduction to Euclid's Les eléments, i (Paris, 1990).

Valla Giorgio , De expetentis et fugentibus rebus (Venice, 1502).

Theon of Smyrna, Elementorum libri XIII (Venice, 1509).

Pacioli Luca , Euclidis Megarensis philosophi acutissimi mathematicorumque omnium sine controversia principis opera a Campano interprete fidissimo tralata (Venice, 1509).

Euclides Megarensis Geometricum elementorum libri XV (Paris, 1516). Grynaeus Simon , Euclides … Geometricum elementorum (Basel, 1533), included Proclus's commentaries.

Euclides Megarensis Geometricum elementorum libri (Basel, 1539).

Fontana Nicolo , Euclide Megarense solo introduttore delle scienze mathematiche (Venice, 1547). Other translations into vernacular languages were published in the sixteenth century: In French by Pierre Forcadel in 1564, and in English by Henry Billingsley in 1570. In 1572, the next important step was a Latin translation from the Greek made by Federico Commandino, printed in Pesaro; this version differs strikingly from the texts based on Campanus. Commandino translated it into Italian three years later under the title De gli elementi d'Euclide libri quindici (Urbino, 1576).

Cagnani Angelo , Euclides I quindici libri degli elementi di Greco tradotti in lingua Toscana (Rome, 1545).

Bachot , Le gouvernail (ref. 17), 8–9: “Estant donné un cercle par le moyen de son diamètre descrire le Pentagone”.

Bachot , La practique de geometrie (manuscript addition to Le gouvernail), f. 4.

Cf. Chuquet Nicolas , La géométrie: Première géométrie algébrique en langue française, 1484, ed. by l'Huilier H. (Paris, 1979). See also, Bovelles Charles , La géométrie française (Paris, 1511) and Livre singulier et utile, touchant l'art et pratique de géométrie (Paris, 1542). For a comment on Chuquet and Bovelles's mathematics see Margolin Jean-Claude , “L'enseignement des mathématiques en France (1540–1570): Bovelles, Fine, Peletier, Ramus”, in French Renaissance studies (1540–1570), ed. by Sharratt P. (Edinburgh, 1976). For examples of the Italian tradition, see for instance the anonymous Florentine text entitled Trattato di geometria pratica. This Trattato di geometria pratica dal codice L.IV.18 della Biblioteca Comunale di Siena, dating from the 1460s, has been edited and commented by Simi Annalisa , of the Università degli Studi di Siena, in the Quaderni del Centro Studi della Matematica Mediaevale, xxi (1993). See also Fibonacci's Practica geometriae (12th century), ed. by Boncompagni Baldassare in the Scritti di Leonardo Pisano matematico del secolo decimoterzo, ii (Rome, 1857–62).

On Hugo of St Victor, see Baron Roger , “Hugonis de Sancto Victore Practica geometriae”, Osiris, xii (1936), 176–224. On the French oral tradition, see the exhibition catalogue “Les bâtisseurs des cathédrales gothiques”, Strasbourg, 3 Sept.–26 Nov. 1989 (Strasbourg, 1989). See also Sheelby Lon R. , “The medieval knowledge of master masons”, Speculum, xlvii (1972), 1972–426, and Jeanne Peiffers's chapter “La géométrie des métiers” in her critical edition of Albrecht Dürer's Geométrie (Paris, 1995). On Chuquet's manuscripts, see also Hervé l'Huillier's introduction to Nicolas Chuquet's La géométrie (Paris, 1979). We know that one copy was kept at the Bibliothèque de Saint Victor, and that another was kept by the mathematician Estienne de la Roche who used it for his Arismeticque et géométrie in 1520 and 1538.

Souffrin Pierre , “La geometria pratica dans les Ludi mathematicarum”, Albertiana, i (1998), 87–104, and his edition of Alberti's Ludi under the title Divertissements mathématiques (Paris, 2002).

See Boysset Bertrand , La siensa de destrar, translated by Motte Magdeleine under the title La siensa de destrar de Bertrand Boysset (1350–1415) (Toulouse, 1988). On Boysset, see also Pierre Portet's internet site at http://boysset.ifrance.com/boysset/introduc.htm. For the Italian tradition, see also de Montepulciano's Orbetano Regole di geometria pratica, edited as Regole di geometria pratica dal manoscritto Moreni 130 (sec. XV) della Biblioteca Riccardiana di Firenze, by Annalisa , Quaderni del Centro Studi della Matematica Medievale, xix (1991).

dal Monte Guidobaldo , Le mechaniche dell'illustrissimo sig. Guido Ubaldo de Marchesi del Monte nelle quali si contiene la vera dottrina di tutti gli istrumenti principali da mover pesi grandissimi con picciola forza (Venice, 1581).

On the self-fashioning of the scholar into a courtier, see Biagioli Mario , Galileo courtier (Chicago, 1991), and on the question of reduction of Euclid to art, see the title of Tartaglia's edition in 1565: Euclide…. Diligentemente rassettato e alla integrità ridotto … con una ampla espositione. To learn more about this concept, see Vérin Hélène , “La réduction en art et la science pratique au XVIe siècle”, in Raisons pratiques (Paris, 1992), 119–44.

Bachot , Le gouvernail (ref. 17), 5: “pour adextrer et façonner à toutes sortes de traces ceux qui se délectent de la fortification”.

Cf. Cuomo , op. cit. (ref. 39), 155–88.

Lanteri , Dialoghi (ref. 8), 5.

For a primary review of these texts relying only on American Libraries resources, see Kapinski L. C. Kokormoor F. W. , “The teaching of Elementary Geometry in the seventeenth century”, Isis, x (1928), 21–32. See also Shelby Lon R. , “The geometrical knowledge of medieval masons”, Speculum, xlvii (1972), 1972–421.

Bartoli Cosimo , Del modo di misurare le distantie, le superficie, i corpi, le piante, le provincie, prospettive secondo le vere regole d'Euclide (Venice, 1564). Vinet Elie , L'arpenterie (Bordeaux, 1577) and Zacarie , “Professeur ès mathématiques”, Traicté d'arithmétique, géométrie, avec l'art d'arpanter et mesurer toutes superficie [sic] de terre… Nouvellement mis en lumière par le sieur Zacarie (Paris, 1618).

Errard Jean , La fortification réduite en art (Paris, 1600), and Marolois Samuel , Geometrie; contenant la théorie et practique d'icelle nécessaire a la fortification. Jadis escrite par Samuel Marolois. Mais du depuis corrigée et la pluspart du discours changé et redigé en meilleur estat par Albert Girard, mathematicien (Amsterdam, 1625).

Ive Paul , The practice of fortification wherein is shewed the manner of fortifying in all sorts of scituations, with the considerations to be used in delining, and making of royal frontiers, skonces, and renforcing of ould walled townes. Compiled in a most easie, and compendious method, by Paule Ive, Gent (London, 1589). Digges Thomas , A geometrical practise, named Pantometria, diuided into three bookes, longimetra, planimetra, and stereometria … (London, 1571), reprinted as A geometrical practical treatize named Pantometria … lately reviewed by the author himselfe and augmented with sundrie “additions”, “diffinitions”, … to open the passage and prepare a way to the understanding of his treatys of martially pyrotechnic and great artillery, hereafter to be published (London, 1591).

There was a massive transfer of skills and works across national boundaries in the Renaissance even though there were national traditions: Oronce Fine's practical geometry, for instance, can be found translated into English in a manuscript of the British Library (Ms Sloane 2102, A short treatise on geometrie) and it is probable that Digges's Pantometria used Fine's Quarré géométrique.

Weber Max , The Protestant ethic and the spirit of capitalism (London, 1974), is one of the first authors to point out that the roots of modern self-discipline lay in army life. Oestreich Gerhard , in Neostoicism and the early modern state (Cambridge, 1982), emphasizes the crucial moment represented by the sixteenth century and the role of Lipsius's ideas on army reform in the Netherlands. In Discipline and punish: The birth of the prison (London, 1977; first published under the title Surveiller et punir (Paris, 1975)), Michel Foucault explained the ways in which, with the beginning of modernity, the “body” was historically disciplined by diverse institutions among which the barracks played a crucial role.

See Reiss Timothy J. “Calculating humans: Mathematics, war and the colonial calculus”, 137–63, and Cahill Patricia , “Killing by computation: Military mathematics, the Elizabethan social body, and Marlowe's Tamburlaine“, in Glimp David (ed.), Arts of calculation: Numerical thought in early-modern Europe (New York, 2004), 165–86.

“sopra una carta, può fare quello che vuole, ed io farò cose sopra una carta, che saranno molto lodate, ed di sorte, che mai si potranno metter in effetto perchè il dissegno inganna, ed può mostrar il falso…”, in Giovan Battista Belluzzi, Trattato delle fortificazioni di terra, ms. written circa 1545 and preserved in the Biblioteca Nazionale in Florence. Fol. 88v, quoted by Horst de la Croix in “The literature on fortification in Renaissance Italy”, Technology and culture, iv (1963), 30–50.

Digges Leonard , An arithmetical militare treatise, named Stratioticos … (London, 1579). Of the same author see also Foure paradoxes (London, 1604) and Pantometria (ref. 87). Barnabe Rich's critique is expounded in Faultes faultes, and nothing else but faultes (London, 1606) and in The fruites of long experience (London, 1604).

Dijksterhuis E. J. , Simon Stevin, science in the Netherlands around 1600 (The Hague, 1970).

Gerbier Balthazar , The first lecture being an introduction to the military architecture or fortifications read publiquely at Sir Balthezar Gerbiers Academy (London, 1649).