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ELS 8 LLIBRES DE PAPPUS. LA COL·LECCIÓ.
Pappus's "Collection"
La Col·lecció de Pappus
consisteix en un recull dels tractats de geometria, astronomia i
mecànica de finals de l´any 300 a.c. i és la última gran obra dels
matemàtics grecs. Aquest manuscrit corona la llibreria Papal de la
seva època i és l´arquetip de les còpies posteriors, de fet és la
més .... de la sisena centúria.
Vat. gr. 218 fols. 39 verso - 40 recto math08a NS.05
Pappus's "Collection" consisting of
supplements
to earlier treatises on geometry, astronomy,
and mechanics, dates from the late third century A.D. and is the last
important work of Greek mathematics.
This manuscript reached the papal library in the
thirteenth century, and is the archetype of all
later copies, of which none is earlier than the sixteenth century.
Vat. gr. 218 fols. 39 verso - 40 recto math08a NS.05 |
Detall de la Col·lecció
Detail of Collection
1589
Liber Tertius
Heath va dir de l´obra de Pappus que:
"Obviously written with the object of reviving the classical Greek
geometry, it covers practically the whole field. It is, however, a
handbook or guide to Greek geometry rather than an encyclopaedia; it was
intended, that is, to be read with the original works (where still
extant) rather than to enable them to be dispensed with."
Llibre I
- El Llibre I i les primeres 13 proposicions del Llibre II estàn perduts.
Llibre II - Was
concerned with very large numbers - powers of myriads.
Llibre III - Aquest Llibre comença
amb un sumari per trobar dues raons proporcionals (a : x = x : y = a : y)
entre dues línies rectes. També es defineixen problemes amb plans, amb
sòlids i problemes linials. Pappus:
Begins with a summary of finding two mean proportionals (a : x
= x : y = a : y) between two straight lines. He also defines plane problems,
solid problems, and linear problems. Pappus :
1- Distinguishes (1) plane problems, solvable with straight edge and
compass
2- Distinguishes (2) solid problems, requiring the conics for solution,
e.g. solving certain cubics.
3- Distinguishes (3) linear problems, problems invoking spirals,
quadratrices, and other higher curves.
4- Gives a constructive theory of means. That is, given any two of the
numbers a, b, c and the type of mean (arithmetic, geometric, or
harmonic), he constructs the third.
5- Describes the solution of the three famous problems of antiquity,
asserts these are not plane problems ~19
century.
6- Treats the trisection problem, giving another solution involving a
hyperbola and a circle.
7- Inscribes the five regular solids in the sphere.
Llibre IV - Covers an extention of theorem of Pythagorus for
parallelograms constructed on the legs of any triangle. Also in Book IV is
material on the Archimedian spiral, including methods of finding area of
one turn -- differs from Archimedes. He also constructs the conchoid of
Nicomedes. In addition, he constructs the quadratix in two different ways,
(1) using a cylindrical helix, and (2) using a right cylinder, the base of
which is an Archimedian spiral.
Llibre V - Reproducció de l´obra de
Zeodorus sobre les figures isoperimètriques. Here we see in the introduction his comments on the sagacity of bees.
Llibre VI - Determina el centre
d´una elipse com la projecció perspectiva d´una circumferència. It is also astronomical in nature. It has been called the "Little
Astronomy''. It covers optics - reflection and refraction.
Llibre VII - El Tractat
d´Analisis és molt important perque
The "Treasury of Analysis"' is very important because it
surveys a great number of works on geometric analysis of loci, nearly all
of which are lost. Features:
The Book begins with a definition of aanalysis and synthesis.
Analysis, then takes that which is sought as if it were admitted and
passes from it through its successive consequences to something which is
admitted as the result of systhesis. Unconditional controvertability
required.
In Synthesis, reversing the process, we take as already done that which
was last arrived at in the analysis and, by arranging in their natural
order as consequences what before were antecedents, and successively
connecting them one with the other, we arrive finally at the construction
of what was sought.
A list of the books forming the "treasury" is included, together with a
short description of their contents.
As an independent contribution Pappus formulated the volume of a solid of
revolution, the result we now call the The Pappus - Guldin Theorem. P.
Guldin (1577-1643)
Most of the remaining of the treatise is collections of lemmas that will
assist the reader's understanding of the original works.
Book I - The
Book I and
first 13 propositions of Book II are missing.
Book II - Was
concerned with very large numbers - powers of myriads.
Book III - Begins with a summary of finding two mean proportionals (a:x
= x:y = a:y) between two straight lines. He also defines plane problems,
solid problems, and linear problems. Pappus :
1- Distinguishes (1) plane problems, solvable with straight edge and
compass
2- Distinguishes (2) solid problems, requiring the conics for solution,
e.g. solving certain cubics.
3- Distinguishes (3) linear problems, problems invoking spirals,
quadratrices, and other higher curves.
4- Gives a constructive theory of means. That is, given any two of the
numbers a, b, c and the type of mean (arithmetic, geometric, or
harmonic), he constructs the third.
5- Describes the solution of the three famous problems of antiquity,
asserts these are not plane problems ~19
century.
6- Treats the trisection problem, giving another solution involving a
hyperbola and a circle.
7- Inscribes the five regular solids in the sphere.
Book IV - Covers an extention of theorem of Pythagorus for
parallelograms constructed on the legs of any triangle. Also in Book IV is
material on the Archimedian spiral, including methods of finding area of
one turn -- differs from Archimedes. He also constructs the conchoid of
Nicomedes. In addition, he constructs the quadratix in two different ways,
(1) using a cylindrical helix, and (2) using a right cylinder, the base of
which is an Archimedian spiral.
Book V - Reproduces the work of Zeodorus on isoperimetric figures.
Here we see in the introduction his comments on the sagacity of bees.
Book VI - Determines the center of an ellipse as a perspective of a
circle. It is also astronomical in nature. It has been called the "Little
Astronomy''. It covers optics - reflection and refraction.
Book VII - The "Treasury of Analysis"' is very important because it
surveys a great number of works on geometric analysis of loci, nearly all
of which are lost. Features:
The Book begins with a definition of aanalysis and synthesis.
Analysis, then takes that which is sought as if it were admitted and
passes from it through its successive consequences to something which is
admitted as the result of systhesis. Unconditional controvertability
required.
In Synthesis, reversing the process, we take as already done that which
was last arrived at in the analysis and, by arranging in their natural
order as consequences what before were antecedents, and successively
connecting them one with the other, we arrive finally at the construction
of what was sought.
A list of the books forming the "treasury" is included, together with a
short description of their contents.
As an independent contribution Pappus formulated the volume of a solid of
revolution, the result we now call the The Pappus - Guldin Theorem. P.
Guldin (1577-1643)
Most of the remaining of the treatise is collections of lemmas that will
assist the reader's understanding of the original works.
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