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=20

16/11/2002

Pedro = Nunes

Um=20 matem=E1tico num pa=EDs de navegantes

=ABOs portugueses ousaram cometer o = grande mar=20 oceano=BB, escrevia Pedro Nunes, =ABdescobriram novas ilhas, = novas=20 terras, novos mares, novos povos: e o que mais =E9: novo = c=E9u, novas=20 estrelas=BB.

Textos = de Nuno=20 Crato=20

=20

ANA BAI=C3O=20
O maior matem=E1tico = da =E9poca=20 das Descobertas nasceu h=E1 500 anos, na cidade de Alc=E1cer = do Sal.=20 Quase nada se sabe sobre a sua origem, pouco se conhece da = sua vida=20 e ignora-se muito da rela=E7=E3o com os cientistas do seu = tempo.=20 Contudo, =E9 poss=EDvel reconstituir o seu trabalho = atrav=E9s das suas=20 publica=E7=F5es e das muitas refer=EAncias que lhe s=E3o = feitas pelos=20 grandes intelectuais da =E9poca. Chamava-se Pedro Nunes e = assim=20 assinava, mas o seu nome aparece muitas vezes na forma = latinizada=20 Petrus Nonius.

Sabe-se apenas o ano de nascimento, n=E3o se = conhece o=20 m=EAs nem o dia. Sabe-se a data exacta da morte, que se = registou em 11=20 de Agosto de 1578. A sua vida, tal como a de Cam=F5es (c. = 1524-1580),=20 coincidiu com o per=EDodo de maior grandeza do nosso pa=EDs. = Pouco ap=F3s=20 a morte destes grandes portugueses, Filipe II de Espanha = ascendeu ao=20 trono de Portugal e o pa=EDs perdeu a independ=EAncia, que = s=F3 viria a=20 recuperar em 1640. Nessa altura, tinha-se j=E1 encerrado a = =E9poca em=20 que Portugal e Espanha dominavam incontestavelmente os = mares. Na=20 Pen=EDnsula Ib=E9rica, a Matem=E1tica e a Ci=EAncia perderam = tamb=E9m a sua=20 maior grandeza.

Ao que parece, Pedro Nunes tinha origens = judaicas,=20 pois Dami=E3o de G=F3is descreveu-o como =ABportugu=EAs de = na=E7=E3o=BB,=20 refer=EAncia habitualmente aplicada a judeus conversos. No = entanto,=20 nada nos escritos de Pedro Nunes mostra alguns sinais de=20 criptojuda=EDsmo e as poucas refer=EAncias religiosas que = neles se=20 encontram s=E3o claramente crist=E3s. N=E3o h=E1 tamb=E9m = quaisquer ind=EDcios=20 de Nunes ter sido alguma vez perseguido - durante toda a = vida, foi=20 protegido por reis e pr=EDncipes. Os seus netos, contudo, = passaram=20 alguns anos a ferros da Inquisi=E7=E3o, acusados de = professarem o=20 juda=EDsmo.

=20

ANA BAI=C3O

R=E9plica do Quadrante de Kynuyn = (Museu de=20 Marinha, em Lisboa) fabricado em 1595 em Londres. =C9 = o =FAnico=20 instrumento sobrevivente da =E9poca dotado do n=F3nio, = descoberto=20 por Est=E1cio dos Reis. O n=F3nio aparece aqui nas 45 = escalas=20 circulares conc=EAntricas marcadas na face do = quadrante

Tal=20 como o matem=E1tico italiano Francisco Maurolico = (1494-1575), Pedro=20 Nunes trabalhou no per=EDodo que medeia entre Regiomontano = (1436-1476)=20 e Vi=E8te (1544-1603). Tal como o astr=F3nomo Tycho Brahe = (1546-1601) e=20 o cosm=F3grafo Crist=F3v=E3o Cl=E1vio (1538-1612), viveu no = per=EDodo=20 interm=E9dio entre Cop=E9rnico (1473-1543) e Galileu = (1564-1642).=20 Entalado entre esses grandes vultos, que marcaram momentos = de=20 viragem na Matem=E1tica e na Ci=EAncia, o trabalho de Pedro = Nunes tem=20 sido menos estudado do que mereceria. Ao tempo, contudo, as = suas=20 contribui=E7=F5es foram plenamente reconhecidas, at=E9 ao = fim da sua=20 vida.

Praticamente no leito de morte, Nunes foi = instado=20 pelos enviados do Papa Greg=F3rio para dar um parecer sobre = a reforma=20 do calend=E1rio, pedido a que j=E1 n=E3o p=F4de = corresponder. Brahe, o maior=20 astr=F3nomo da era pr=E9-telesc=F3pica, experimentou uma das = suas=20 inven=E7=F5es, a que chamou =ABsubtil processo=BB. E = Cl=E1vio, a maior figura=20 do Col=E9gio Romano, o grande centro do saber cat=F3lico da = =E9poca,=20 classificou-o como =ABg=E9nio matem=E1tico supremo=BB.

Pedro Nunes estudou em Lisboa e Salamanca, = onde=20 conheceu Guiomar de Aires, uma crist=E3 castelhana com quem = se casou e=20 de quem teve seis filhos. Depois de obter o bacharelato, foi = contratado como leitor pela Universidade de Lisboa, onde = veio a=20 doutorar-se em Medicina. Na posse desse t=EDtulo = acad=E9mico, obtido na=20 =FAnica =E1rea cient=EDfica da =E9poca que o conferia, Nunes = p=F4de prosseguir=20 as suas investiga=E7=F5es matem=E1ticas numa posi=E7=E3o = confort=E1vel.

Ainda novo, o seu g=E9nio foi notado na = corte. Por=20 decis=E3o do Rei D. Jo=E3o III, foi preceptor dos = pr=EDncipes D. Lu=EDs e D.=20 Henrique, ensinando-lhes L=F3gica, Filosofia, Geometria e = Matem=E1tica.=20 Com 27 anos foi nomeado cosm=F3grafo real, posi=E7=E3o de = grande prest=EDgio=20 no pa=EDs. Duas d=E9cadas mais tarde foi nomeado = cosm=F3grafo-mor.

=20

SCIENCE MUSEUM / SCIENCE & = SOCIETY=20 PICTURE LIBRARY

Instrumento de Sombras de Pedro = Nunes=20 fabricado por Humphrey Cole (1572-1624) em Londres, = existente=20 no Museu de Ci=EAncia desta cidade

Para=20 cumprir as obriga=E7=F5es como cosm=F3grafo e professor dos = pr=EDncipes,=20 Pedro Nunes renunciou =E0 posi=E7=E3o acad=E9mica. Foi de = novo nomeado=20 professor depois da transfer=EAncia da universidade para = Coimbra em=20 1537, vindo a residir o resto da vida nesta cidade. = Jubilou-se em=20 1562 e morreu em Coimbra em 1578. Era frequentemente chamado = a=20 Lisboa, para aconselhar e examinar os pilotos, para ajudar = na=20 cartografia das terras e mares ent=E3o descobertos e para = tratar de=20 assuntos cient=EDficos de interesse para o reino.

Pouco ap=F3s a nomea=E7=E3o como professor = em Coimbra,=20 escreveu =ABDe erratis Orontii Finaei=BB, a obra mais = pol=E9mica que=20 alguma vez assinou. Tratava-se de uma refuta=E7=E3o das = pretens=F5es do=20 matem=E1tico do Col=E9gio Real de Paris, Oronce Fine = (1494-1555), que=20 afirmava ter resolvido os tr=EAs problemas geom=E9tricos = cl=E1ssicos da=20 Geometria grega: a trissec=E7=E3o de qualquer =E2ngulo, a = duplica=E7=E3o do=20 cubo e a quadratura do c=EDrculo, com recurso apenas a regra = n=E3o=20 graduada e compasso. Pedro Nunes n=E3o provou a = impossibilidade de=20 resolu=E7=E3o desses problemas, pois os instrumentos = matem=E1ticos da=20 =E9poca n=E3o permitiam fazer tal demonstra=E7=E3o, s=F3 = conseguida no s=E9culo=20 XIX. Contudo, a refuta=E7=E3o das =ABprovas=BB apresentadas = pelo matem=E1tico=20 franc=EAs foi de um rigor impec=E1vel. =C9 significativo que = o=20 rec=E9m-chegado matem=E1tico de Coimbra tivesse sido o = primeiro na=20 Europa a defrontar o professor franc=EAs, mostrando assim a=20 superioridade do seu dom=EDnio da Geometria.

Os trabalhos mais importantes de Pedro = Nunes, no=20 entanto, s=E3o anteriores a =ABDe erratis=BB e =E0 = nomea=E7=E3o para a=20 Universidade de Coimbra. A primeira obra impressa do = matem=E1tico, o=20 =ABTratado da Esfera=BB, foi publicada em 1537 em Lisboa. = =C9 composta por=20 v=E1rios trabalhos, como era costume na =E9poca. Nele se = incluem=20 tradu=E7=F5es de obras cl=E1ssicas, enriquecidas com = coment=E1rios e adi=E7=F5es=20 que denotam um conhecimento profund=EDssimo da dif=EDcil = cosmografia da=20 =E9poca. Nele se incluem tamb=E9m dois pequenos trabalhos = originais em=20 que Nunes avan=E7a a sua teoria da curva loxodr=F3mica, com = a qual=20 discute a navega=E7=E3o pelo caminho mais curto e a de = direc=E7=E3o cardeal=20 constante. A curva loxodr=F3mica constitui talvez a = descoberta mais=20 importante de Pedro Nunes e aquela que este mais = considerava. Como=20 acontece com muitos matem=E1ticos, =E9 tamb=E9m a sua = primeira descoberta=20 de vulto, pelo menos se nos guiarmos pela data da = publica=E7=E3o. Volta=20 a ser discutida em 1566, na recolha das suas obras = publicadas em=20 Basileia. Se se tratasse apenas da descoberta de uma curva=20 matem=E1tica, o tema poderia ser interessante e ter = repercuss=F5es no=20 desenvolvimento desta disciplina, como teve, mas tratou-se = de muito=20 mais do que isso. Foi a resolu=E7=E3o geom=E9trica de um = problema=20 essencial =E0 navega=E7=E3o e mudou a forma de cartografar = os oceanos e=20 representar o mundo.

=20

NUNO BOTELHO

Est=E1tua de Pedro Nunes na sua = cidade natal,=20 Alc=E1cer do Sal

A=20 publica=E7=E3o seguinte de Pedro Nunes =E9 =ABDe = Crepusculis=BB, habitualmente=20 considerado o seu trabalho mais importante. Nessa obra, = publicada em=20 1542 em Lisboa, resolve um problema de geometria esf=E9rica = muito=20 dif=EDcil, o da determina=E7=E3o rigorosa da dura=E7=E3o dos = crep=FAsculos para=20 dada latitude e dado dia do ano.

A cria=E7=E3o que tornou Pedro Nunes mais = conhecido foi o=20 n=F3nio, tamb=E9m proposto no =ABDe Crepusculis=BB. Trata-se = de um sistema=20 de conjuga=E7=E3o de escalas que permite obter medidas mais = precisas que=20 qualquer das escalas individuais. Com esse sistema, Nunes = avan=E7a uma=20 solu=E7=E3o para um problema premente na =E9poca, o problema = da medida=20 rigorosa dos =E2ngulos de altura de astros. O n=F3nio surge = como um=20 sistema de 45 escalas num astrol=E1bio, mas veio mais tarde = a ser=20 aplicado a outros instrumentos de medida, sendo hoje comum = nas=20 craveiras usadas para medir comprimentos.

O n=F3nio veio a sofrer diversos = aperfei=E7oamentos,=20 cabendo ao franc=EAs Pierre Vernier (1584-1638) a gl=F3ria = de ter=20 proposto, em 1631, a solu=E7=E3o mais pr=E1tica, que reduz o = n=F3nio a duas=20 escalas, sendo uma delas fixa e outra m=F3vel. O pr=F3prio = Vernier dizia=20 que o seu invento era um n=F3nio aperfei=E7oado e, durante = muito tempo,=20 foi conhecido com o nome de =ABnonius=BB, mesmo em Fran=E7a. = No s=E9culo=20 XVIII, contudo, os franceses passaram a utilizar a = designa=E7=E3o=20 =ABvernier=BB, que se generalizou em quase todo o mundo. = Vendo hoje os=20 dois sistemas, =E9 dif=EDcil perceber tratar-se de duas = formas de=20 aplicar o mesmo princ=EDpio, mas estudando a sua = evolu=E7=E3o, nota-se a=20 continuidade da ideia do nosso matem=E1tico.

Pedro Nunes veio ainda a produzir trabalhos = not=E1veis=20 de Geometria e de =C1lgebra, assim como a apresentar = propostas=20 inovadoras para resolver v=E1rios problemas de navega=E7=E3o = do seu tempo.=20 Algumas das suas ideias n=E3o tiveram aplicabilidade, outras = acabaram=20 por ser =FAteis, embora nem sempre tenham sido imediatamente = compreendidas. Sabe-se, por exemplo, que nem sempre as suas = rela=E7=F5es=20 com os homens do mar foram as mais produtivas. Segundo o = pr=F3prio=20 escreveu, =ABsei qu=E3o mal sofrem os pilotos que fale na = =CDndia quem=20 nunca foi nela, e pratique no mar quem nele n=E3o entrou=BB. = Mas Pedro=20 Nunes era um matem=E1tico e um cosm=F3grafo, os seus = trabalhos de=20 navega=E7=E3o astron=F3mica interessavam-lhe, sobretudo, = pelos problemas=20 te=F3ricos que neles reconhecia e com que se inspirava. = Sabia que os=20 portugueses tinham descoberto =ABnovas ilhas, novas terras, = novos=20 mares, novos povos=BB, mas Pedro Nunes olhava para cima e = havia algo=20 que mais o interessava. Disse-o logo a seguir: =ABE o que = mais =E9: novo=20 c=E9u, novas estrelas=BB.


	O n=F3nio

No s=E9culo XVI, a navega=E7=E3o = estava dependente dos=20 astrol=E1bios e de outros instrumentos de medida de = alturas. No=20 entanto, a precis=E3o obtida por esses instrumentos = estava=20 limitada pelo rigor da escala graduada. Essa escala = tinha,=20 habitualmente, a unidade m=EDnima de um grau, que = podia ser=20 subdividida em meios graus, mas que n=E3o podia ir = muito mais=20 longe, pois as pr=F3prias marcas feitas no metal t=EAm = uma=20 espessura e come=E7am a sobrepor-se se estiverem = demasiado perto=20 umas das outras.

Nunes tem ent=E3o a ideia de marcar no = astrol=E1bio=20 v=E1rias escalas com diferentes unidades. Para = perceber a sua=20 ideia basta pensar na subdivis=E3o de um =E2ngulo = recto, que=20 corresponde a um quadrante e que =E9 o suficiente para = a medida=20 de alturas astron=F3micas. No sistema idealizado por = Nunes,=20 havia 45 escalas conc=EAntricas marcadas no quadrante, = como=20 acontece no instrumento de Kynuyn de que se mostra a = r=E9plica=20 existente no Museu de Marinha (p=E1gina =E0 direita). = A escala=20 exterior dividia o =E2ngulo recto em 90 partes, = portanto com=20 unidades de um grau. Interiormente a essa, estava = graduada uma=20 escala que dividia o =E2ngulo em 89 partes, portanto = com=20 unidades de 90/89 de grau. Logo abaixo graduava-se uma = outra=20 que dividia o =E2ngulo recto em 88 partes, e assim=20 sucessivamente at=E9 chegar a uma escala que dividia o = quadrante=20 em 46 partes. No n=F3nio havia pois 45 escalas = conc=EAntricas,=20 cada uma com menos uma subdivis=E3o que a que lhe era=20 exterior.

Para medir a altura angular de um = astro, o=20 observador ajustava a mira do astrol=E1bio ou = quadrante, a=20 chamada mediclina, de forma a alinh=E1-la com a = direc=E7=E3o do=20 astro. Media depois esse =E2ngulo, mas n=E3o estava = limitado=20 apenas a uma escala, como nos instrumentos = tradicionais. A=20 mediclina passava por todas as escalas conc=EAntricas = e o=20 observador escolhia aquela onde existia uma marca que = melhor=20 coincidia com a posi=E7=E3o da mediclina. Suponhamos, = como o fez=20 Pedro Nunes no =ABDe Crepusculis=BB, que a marca que = melhor=20 coincidia com a posi=E7=E3o da mediclina se encontrava = na quarta=20 escala a contar do exterior, aquela que dividia o = quadrante em=20 87 partes. E imaginemos que coincidia com a marca 30 = dessa=20 escala. Sendo assim, o =E2ngulo medido corresponde a = 30/87 do=20 quadrante. Fazendo-se as contas, a frac=E7=E3o 30/87 = corresponde=20 aproximadamente a 31 2' 4'', valor que se obt=E9m com = maior=20 precis=E3o que a de qualquer das escalas individuais = usadas.

A proposta de Pedro Nunes foi estudada = em toda a=20 Europa. O maior astr=F3nomo da =E9poca, o = dinamarqu=EAs Tycho Brahe=20 (1546-1601), construiu v=E1rios quadrantes graduados = com o=20 n=F3nio. =ABRecorri ao subtil processo que Nunes = apresenta=BB, diz=20 Tycho, =ABe tornei-o mais preciso, aumentando o = n=FAmero de=20 subdivis=F5es e calculando tabelas=BB. No entanto, o = astr=F3nomo=20 acabou por confessar que o instrumento n=E3o lhe = permitia obter=20 a precis=E3o que desejava. Procuraram-se ent=E3o = solu=E7=F5es pr=E1ticas=20 que melhorassem o sistema.

O alem=E3o Jacob Curtius, figura = influente na=20 Europa central e no Vaticano, prop=F4s uma = altera=E7=E3o ao n=F3nio,=20 introduzindo escalas graduadas de forma mais simples. = O=20 matem=E1tico jesu=EDta Crist=F3v=E3o Cl=E1vio = (1538-1616) utilizou a=20 ideia de Curtius e simplificou-a ainda mais, limitando = o n=F3nio=20 a duas escalas. Na sua =ABGeometria practica=BB, = publicada em Roma=20 em 1604, prop=F4s que uma escala tivesse unidades de = um grau e=20 outra unidades de um grau e um sexag=E9simo de grau, = isto =E9, um=20 grau e um minuto (1 + 1/60 =3D 1 1').

Com esse processo, podem obter-se = directamente=20 medidas de minutos de grau por compara=E7=E3o das duas = escalas. A=20 primeira marca da escala principal e a primeira da = secund=E1ria=20 distam uma da outra de um minuto de grau. As segundas = marcas=20 de cada escala distam uma da outra de dois minutos de = grau, e=20 assim sucessivamente. Segundo o processo de Cl=E1vio, = come=E7ava=20 por se medir o =E2ngulo atrav=E9s da escala principal. = Essa medida=20 dava um certo n=FAmero de graus e, naturalmente, um = pequeno=20 excesso. Para medir essa frac=E7=E3o de grau residual, = Cl=E1vio=20 propunha que se utilizasse um compasso e que o excesso = se=20 comparasse com as diferen=E7as entre as marcas das = duas escalas.=20 Se esse excesso fosse igual =E0 diferen=E7a entre as = d=E9cimas=20 marcas da escala principal e auxiliar, por exemplo, = isso=20 significava que o excesso media 10 minutos de grau. =

O processo =E9 inteligente, mas =E9 = ainda pouco=20 pr=E1tico. Foi necess=E1rio esperar mais trinta anos = para que=20 algu=E9m aparecesse com um m=E9todo expedito de = transportar=20 medidas de uma escala para a outra. Esse algu=E9m foi = o franc=EAs=20 Pierre Vernier (1584-1638), que em 1631 dotou o = instrumento de=20 uma escala m=F3vel em que deslocava o arco auxiliar,=20 possibilitando a medida directa do =E2ngulo residual = imaginado=20 por Cl=E1vio.

O invento de Vernier teve grande = sucesso e=20 difundiu-se entre os instrumentistas. Adaptou-se a = sextantes,=20 craveiras e outros instrumentos ainda hoje muito = usados. =C9=20 essa vers=E3o do n=F3nio que hoje se = utiliza.

=
ESQUEMA do=20 N=F3nio tal como aparece no =ABDe Crepusculis=BB e que = Pedro Nunes=20 explica nos seguintes termos: =ABLevantaremos o = astrol=E1bio acima=20 dos olhos (...) e dirigiremos a parte ab para a = estrela,=20 andando levemente com a medeclina para cima e para = baixo at=E9 a=20 enfiarmos com a vista atrav=E9s dos dois orif=EDcios. = como rara=20 ser=E1 a vez em que a medeclina se sobrep=F5e aos = ditos quadrantes=20 sem cortar algum deles segundo o tra=E7o de uma = divis=E3o,=20 tomaremos nota do n=FAmero das partes inteiras, que a = posi=E7=E3o=20 cortada tem, e do n=FAmero em que todo o quadrante = estiver=20 dividido, e pela sabida regra dos n=FAmeros = proporcionais=20 converteremos estas partes em nonag=E9simas partes do = quadrante,=20 as quais vulgarmente se chamam graus.=BB

=
Na foto =E0=20 direita, modelo did=E1ctico de craveira graduada com = n=F3nio=20 pertencente ao laborat=F3rio de f=EDsica do Col=E9gio = Militar.=20 Trata-se da vers=E3o de n=F3nio proposta por Vernier, = adaptada =E0=20 medida de comprimentos. O comprimento aqui marcado = pela=20 craveira =E9 de 5 unidades e 6 d=E9cimas. As unidades = s=E3o lidas na=20 escala principal, mas v=EA-se que o comprimento = ultrapassa as 5.=20 Cada marca da escala m=F3vel (n=F3nio) mede 9/10 da = unidade da=20 r=E9gua fixa. Procura-se no n=F3nio a marca que melhor = coincide=20 com uma marca da escala principal. Neste caso =E9 a = sexta, o que=20 indica que o excesso sobre as 5 unidades =E9 de 6=20 d=E9cimas.


	A curva mais direita

=20

Gravura de Escher: linhas=20 loxodr=F3micas convergindo para os p=F3los

A=20 descoberta da curva loxodr=F3mica =E9 um dos maiores = feitos de=20 Pedro Nunes. Segundo o pr=F3prio escreveu, foi levado = a=20 descobrir essa curva para responder a problemas = pr=E1ticos de=20 navega=E7=E3o que lhe foram levantados por Martim = Afonso de Sousa,=20 fundador das primeiras col=F3nias que Portugal teve no = Brasil.=20 Querendo vir do Rio da Prata para Lisboa, o navegador = desejava=20 perceber qual era a direc=E7=E3o exacta em que deveria = viajar.

O g=E9nio de Pedro Nunes consiste em = ter=20 distinguido claramente duas poss=EDveis traject=F3rias = para um=20 navio no mar alto. Uma seria a traject=F3ria de = dist=E2ncia m=EDnima=20 entre dois pontos, correspondendo a um arco de = c=EDrculo m=E1ximo=20 - =E9 a chamada ortodromia. Outra seria a = traject=F3ria seguida=20 por um barco que mantivesse sempre a mesma = orienta=E7=E3o em=20 rela=E7=E3o aos pontos cardeais - =E9 a chamada linha = de rumo, mais=20 tarde conhecida como loxodromia. As duas = traject=F3rias s=F3=20 coincidem: quando se viaja ao longo do equador, de = leste para=20 oeste ou de oeste para leste, ou quando se viaja ao = longo de=20 um meridiano, de norte para sul ou de sul para norte. = Em todos=20 os outros casos, elas s=E3o diferentes. Para seguir a=20 traject=F3ria m=EDnima entre Lisboa e Nova Iorque, por = exemplo, os=20 avi=F5es n=E3o seguem uma linha de rumo, mudam = constantemente de=20 orienta=E7=E3o cardeal, para poderem seguir um arco de = c=EDrculo=20 m=E1ximo. Nesse trajecto, come=E7am por seguir a = direc=E7=E3o=20 aproximada de noroeste, para, a meio do Atl=E2ntico, = passarem a=20 seguir aproximadamente para sudoeste. =C9 assim que = conseguem=20 seguir o caminho mais curto.

Ao descobrir a linha de rumo, Pedro = Nunes=20 estudou tamb=E9m a traject=F3ria de um navio que, = hipoteticamente,=20 num planeta completamente coberto de =E1gua, seguisse = sempre uma=20 mesma direc=E7=E3o cardeal. Se essa direc=E7=E3o fosse = interm=E9dia=20 entre a norte-sul e a este-oeste, o navio n=E3o = acabaria por=20 regressar ao mesmo lugar, como na altura se pensava, = mas=20 seguiria uma espiral infinita, aproximando-se de um = dos p=F3los=20 e descrevendo um n=FAmero infinito de voltas em seu = torno. Essa=20 curva, que =ABn=E3o =E9 c=EDrculo nem linha = direita=BB, como dizia o=20 nosso matem=E1tico, veio a ser conhecida como espiral=20 loxodr=F3mica e a ser estudada analiticamente por = Gottfried=20 Leibniz (1646-1716) e por outros matem=E1ticos = posteriores.

=
Al=E9m de=20 mostrar aos navegadores o caminho que se segue quando = se toma=20 um rumo cardeal constante, as loxodr=F3micas de Pedro = Nunes=20 tiveram um efeito decisivo na nossa vis=E3o do mundo, = pois=20 determinaram a maneira como os mapas come=E7aram a ser = feitos e=20 criaram a vis=E3o que hoje temos dos continentes. O = matem=E1tico=20 percebeu que os antigos mapas portulanos tinham de ser = refeitos, mas foi Gerardus Mercator (1512-1594) quem = haveria=20 de originar uma revolu=E7=E3o em cartografia, com base = na=20 descoberta de Pedro Nunes.

O grande problema da constru=E7=E3o de = mapas planos=20 =E9 o aparecimento de deforma=E7=F5es inevit=E1veis. = Tal como ao=20 calcar uma casca de laranja sobre uma mesa, esta se = fractura e=20 deforma, assim os fazedores de mapas t=EAm de deformar = a=20 geometria do globo para poder reproduzir uma = superf=EDcie=20 esf=E9rica sobre um plano. As deforma=E7=F5es n=E3o = s=E3o graves quando=20 se trata de =E1reas reduzidas. Mas o mundo tinha-se = tornado=20 maior com as viagens dos Descobrimentos e come=E7avam = a fazer-se=20 mapas de continentes e oceanos, que eram necess=E1rios = para as=20 navega=E7=F5es.

Mercator, que naturalmente conhecia os = resultados de Pedro Nunes pois estes estavam muito = divulgados=20 na Europa, resolveu construir um mapa que servisse=20 directamente os navegadores. Como primeiro princ=EDpio = decidiu=20 que o mapa manteria uma rede, em que as linhas de = igual=20 latitude seriam todas paralelas ao equador e = perpendiculares=20 aos meridianos, e em que os meridianos seriam todos = paralelos=20 entre si. Neste aspecto, limitou-se a seguir as = indica=E7=F5es de=20 Pedro Nunes, que propunha altera=E7=F5es aos =ABmapas=20 rectangulares=BB, isto =E9, aos mapas com = quadr=EDculas marcadas por=20 paralelos e meridianos. Como segundo princ=EDpio, = Mercator=20 decidiu que as linhas de rumo apareceriam como rectas, = limitando-se, tamb=E9m neste aspecto, a seguir uma = sugest=E3o de=20 Pedro Nunes. Para o conseguir, Mercator aumentou=20 progressivamente as dist=E2ncias entre os paralelos = =E0 medida que=20 a latitude se aproxima dos p=F3los. Tamb=E9m aqui = seguiu uma=20 sugest=E3o do matem=E1tico portugu=EAs, que falava da = necessidade de=20 distorcer a escala de latitudes para poder construir = um mapa=20 onde as loxodromias fossem linhas rectas. Assim nasceu = a=20 chamada projec=E7=E3o de Mercator, ainda hoje a mais = conhecida e=20 mais utilizada.


	Saber mais

=
Para=20 conhecer Pedro Nunes, o melhor ponto de partida deve = ser o=20 portal http://www.pedronunes.pt/, constru=EDdo = pelo=20 Ci=EAncia Viva. A=ED se encontram not=EDcias de = actividades, imagens=20 e muitos artigos interessantes, incluindo = reprodu=E7=E3o de v=E1rios=20 artigos cl=E1ssicos fundamentais. Outro local = importante da=20 Internet =E9 o organizado pelo prof. Henrique Leit=E3o = em=20 scientia.artenumerica.org . Outro com interesse =E9 = =ABCi=EAncia em=20 Portugal: Personagens e Epis=F3dios=BB, mantido em www.instituto-camoes.pt/cvc/ciencia.

A =ABGazeta de Matem=E1tica=BB, n.=BA = 143, de Junho=20 deste ano, tem artigos de v=E1rios especialistas = nacionais e=20 internacionais sobre a ci=EAncia no s=E9culo XVI. No = seu conjunto,=20 esta publica=E7=E3o da Sociedade Portuguesa de = Matem=E1tica (tel.=20 217939785) apresenta uma panor=E2mica dos aspectos = essenciais=20 dos trabalhos de Nunes. A revista =ABOceanos=BB, n.=BA = 49, acabada=20 de sair, tem tamb=E9m artigos diversos e profusamente=20 ilustrados, concentrando-se no contexto hist=F3rico = das=20 contribui=E7=F5es do matem=E1tico.

=
No fim de=20 semana passado cerca de quatro centenas de pessoas=20 participaram na confer=EAncia internacional =ABPedro = Nunes e a=20 Ci=EAncia do seu tempo=BB, organizada pela Sociedade = Portuguesa de=20 Matem=E1tica na Universidade de Coimbra. Dia 13 de = Dezembro=20 pelas 14h30, a mesma sociedade realiza no ISEC, em = Coimbra,=20 uma outra sess=E3o em que ser=E3o discutidas as = principais=20 descobertas de Pedro Nunes.

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.tit-evento-assinatura-cartaz { FONT-SIZE: 10px; LINE-HEIGHT: 10px; FONT-FAMILY: "Arial Black", Arial, = Helvetica, Sans-serif; LETTER-SPACING: -1px } .legenda-cartaz { FONT-WEIGHT: bold; FONT-SIZE: 11px; LINE-HEIGHT: 12px; FONT-FAMILY: = Arial, Helvetica, sans-serif } .orel-cartaz { FONT-SIZE: 10px; COLOR: #ffffff; LINE-HEIGHT: 11px; FONT-FAMILY: "Arial = Black", Arial, Helvetica, Sans-serif; BACKGROUND-COLOR: #003366 } .thead-cartaz { FONT-SIZE: 10px; COLOR: #ffffff; LINE-HEIGHT: 11px; FONT-FAMILY: "Arial = Black", Arial, Helvetica, Sans-serif; BACKGROUND-COLOR: #003366 } .texto-caixas-cartaz { FONT-SIZE: 10px; FONT-FAMILY: Arial, Helvetica, sans-serif } .tit-cartaz { FONT-SIZE: 24px; LINE-HEIGHT: 20px; FONT-FAMILY: Arial, Helvetica, = sans-serif } .tit-cartaz-arti { FONT-SIZE: 28px; LINE-HEIGHT: 22px; FONT-FAMILY: Arial, Helvetica, = sans-serif } .tit-cartaz-class { FONT-SIZE: 11px; FONT-FAMILY: "Arial Black", Arial, Helvetica, = Sans-serif; LETTER-SPACING: -1px } .caixa-pesquisa { FONT-SIZE: 11px; VERTICAL-ALIGN: middle; LINE-HEIGHT: 11pt; = FONT-FAMILY: "Arial Black", Arial, Helvetica, Sans-serif; = LETTER-SPACING: -1px } .ablack-9-9 { FONT-SIZE: 9px; COLOR: #000000; LINE-HEIGHT: 9px; FONT-FAMILY: "Arial = Black", Arial, Helvetica, Sans-serif } .ablack-10-b2 { FONT-SIZE: 10px; COLOR: #003366; FONT-FAMILY: "Arial Black", Arial, = Helvetica, Sans-serif } .arial-10-10 { FONT-SIZE: 10px; LINE-HEIGHT: 10px; FONT-FAMILY: Arial, Helvetica, = sans-serif } .arial-11-11 { FONT-SIZE: 12px; LINE-HEIGHT: 12px; FONT-FAMILY: Arial, Helvetica, = sans-serif } .arial-24-20 { FONT-SIZE: 24px; LINE-HEIGHT: 20px; FONT-FAMILY: Arial, Helvetica, = sans-serif } .ablack-9-9-w-k { FONT-SIZE: 9px; COLOR: #ffffff; LINE-HEIGHT: 9px; FONT-FAMILY: "Arial = Black", Arial, Helvetica, Sans-serif; BACKGROUND-COLOR: #000000 } .ablack-10-b { FONT-SIZE: 10px; COLOR: #336699; FONT-FAMILY: "Arial Black", Arial, = Helvetica, Sans-serif } .ablack-10-w { FONT-SIZE: 10px; COLOR: #ffffff; FONT-FAMILY: "Arial Black", Arial, = Helvetica, Sans-serif } .ablack-10-10-r { FONT-SIZE: 10px; COLOR: #cc0033; LINE-HEIGHT: 10px; FONT-FAMILY: "Arial = Black", Arial, Helvetica, Sans-serif } .ablack-10-11-w-b { FONT-SIZE: 10px; COLOR: #ffffff; LINE-HEIGHT: 11px; FONT-FAMILY: "Arial = Black", Arial, Helvetica, Sans-serif; BACKGROUND-COLOR: #003366 } .ablack-11-9-1 { FONT-SIZE: 11px; VERTICAL-ALIGN: middle; FONT-FAMILY: "Arial Black", = Arial, Helvetica, Sans-serif; LETTER-SPACING: -1px } .ablack-11-11-1 { FONT-SIZE: 11px; LINE-HEIGHT: 11px; FONT-FAMILY: "Arial Black", Arial, = Helvetica, Sans-serif; LETTER-SPACING: -1px } .ablack-11-11-1-gy { FONT-SIZE: 11px; COLOR: #999999; LINE-HEIGHT: 11px; FONT-FAMILY: "Arial = Black", Arial, Helvetica, Sans-serif; LETTER-SPACING: -1px } .ablack-11-10-b { FONT-SIZE: 11px; COLOR: #003366; LINE-HEIGHT: 10pt; FONT-FAMILY: "Arial = Black", Arial, Helvetica, sans-serif } .ablack-12-10 { FONT-SIZE: 12px; LINE-HEIGHT: 10pt; FONT-FAMILY: "Arial Black", Arial, = Helvetica, Sans-serif } .ablack-12-10-b { FONT-SIZE: 12px; COLOR: #003366; LINE-HEIGHT: 10pt; FONT-FAMILY: "Arial = Black", Arial, Helvetica, sans-serif } .ablack-12-16-caps { FONT-SIZE: 12px; TEXT-TRANSFORM: uppercase; LINE-HEIGHT: 16px; = FONT-FAMILY: "Arial Black", Arial, Helvetica, Sans-serif } .ablack-15-15-1-w { FONT-SIZE: 15px; COLOR: #ffffff; LINE-HEIGHT: 15px; FONT-FAMILY: "Arial = Black", Arial, Helvetica, Sans-serif; LETTER-SPACING: -1px } .ablack-12-12-1-w-b { FONT-SIZE: 12px; COLOR: #ffffff; LINE-HEIGHT: 15px; FONT-FAMILY: "Arial = Black", Arial, Helvetica, Sans-serif; LETTER-SPACING: 1px; = BACKGROUND-COLOR: #003366 } .ablack-15-15-1-w-b { FONT-SIZE: 15px; COLOR: #ffffff; LINE-HEIGHT: 17px; FONT-FAMILY: "Arial = Black", Arial, Helvetica, Sans-serif; LETTER-SPACING: -1px; = BACKGROUND-COLOR: #003366 } .ablack-50-40 { FONT-SIZE: 50px; LINE-HEIGHT: 40px; FONT-FAMILY: "Arial Black", Arial, = Helvetica, Sans-serif } .times-10-10 { FONT-SIZE: 10px; LINE-HEIGHT: 10px; FONT-FAMILY: "Times New Roman", = Times, serif } .times-b-10-10-caps { FONT-WEIGHT: bold; FONT-SIZE: 10pt; TEXT-TRANSFORM: uppercase; = LINE-HEIGHT: 10px; FONT-FAMILY: "Times New Roman", Times, serif } .times-b-12-10 { FONT-WEIGHT: bold; FONT-SIZE: 12px; LINE-HEIGHT: 10px; FONT-FAMILY: = "Times New Roman", Times, serif } .times-b-12-12-b { FONT-WEIGHT: bold; FONT-SIZE: 12px; COLOR: #336699; LINE-HEIGHT: 12px; = FONT-FAMILY: "Times New Roman", Times, serif } .times-b-12-12-gy { FONT-WEIGHT: bold; FONT-SIZE: 12px; COLOR: #999999; LINE-HEIGHT: 12px; = FONT-FAMILY: "Times New Roman", Times, serif } .times-13-13 { FONT-SIZE: 13px; LINE-HEIGHT: 13px; FONT-FAMILY: "Times New Roman", = Times, serif } .times-b-13-13-caps { FONT-WEIGHT: bold; FONT-SIZE: 13px; TEXT-TRANSFORM: uppercase; = LINE-HEIGHT: 13px; FONT-FAMILY: "Times New Roman", Times, serif } .arial-15-15 { FONT-SIZE: 15px; LINE-HEIGHT: 15px; FONT-FAMILY: Arial, Helvetica, = sans-serif } .arial-15-15-w-b { FONT-SIZE: 15px; COLOR: #ffffff; LINE-HEIGHT: 15px; FONT-FAMILY: Arial, = Helvetica, sans-serif; BACKGROUND-COLOR: #336699 } .times-16-11 { FONT-SIZE: 16px; LINE-HEIGHT: 11px; FONT-FAMILY: "Times New Roman", = Times, serif } .times-b-17-13 { FONT-WEIGHT: bold; FONT-SIZE: 17px; LINE-HEIGHT: 13px; FONT-FAMILY: = "Times New Roman", Times, serif } .times-18-12-w { FONT-SIZE: 18px; COLOR: #ffffff; LINE-HEIGHT: 12px; FONT-FAMILY: "Times = New Roman", Times, serif } .times-24-16-w { FONT-SIZE: 24px; COLOR: #ffffff; LINE-HEIGHT: 16px; FONT-FAMILY: "Times = New Roman", Times, serif } .times-28-28 { FONT-SIZE: 28px; LINE-HEIGHT: 28px; FONT-FAMILY: "Times New Roman", = Times, serif } .times-36-36 { FONT-SIZE: 36px; LINE-HEIGHT: 36px; FONT-FAMILY: "Times New Roman", = Times, serif } .times-i-12-12 { FONT-SIZE: 12px; LINE-HEIGHT: 12px; FONT-STYLE: italic; FONT-FAMILY: = "Times New Roman", Times, serif } .abril-high { FONT-SIZE: 75%; COLOR: #933d00; LINE-HEIGHT: 140%; FONT-FAMILY: "Arial = Black", Arial, Helvetica, Sans-serif } .abril-link { FONT-SIZE: 75%; COLOR: #003366; LINE-HEIGHT: 140%; FONT-FAMILY: "Arial = Black", Arial, Helvetica, Sans-serif } .regtextopq { FONT-WEIGHT: normal; FONT-SIZE: 11px; COLOR: #000000; FONT-FAMILY: = Arial, Helvetica; TEXT-ALIGN: left } .regtexto { FONT-WEIGHT: normal; FONT-SIZE: 12px; COLOR: #000000; LINE-HEIGHT: = 14pt; FONT-FAMILY: Arial, Helvetica; TEXT-ALIGN: left; TEXT-DECORATION: = none } .regbulletnumero12red { FONT-WEIGHT: bold; FONT-SIZE: 12px; COLOR: #990000; LINE-HEIGHT: 16px; = FONT-FAMILY: Verdana, Arial, Helvetica, sans-serif } ------=_NextPart_000_0000_01C2CF0D.BEBBD320--