490246
[Translation: ]
[Translation: ]
[Commentary:
Propositions on triangular numbers;
this sheet was probably placed here deliberately because of the importance of triangular numbers
in Harriot's method of interpolation
Beery and Stedall .)
The refrence to Viète is to his Variorum responsorum liber VII, Chapter IX, Proposition 14 (Viete 1593d, Chapter 9, Prop .
Propositio XIV.
Si fuerint lineae quotcunque æqualiter sese excedentes, fit autem prima excessui æqualis: octuplum ejus quod fit sub minima & composita ex omibus, adjunctum minimae quadrato, æquatur quadrato compositæ ex minima & extrema
If there are any number of lines exceeding each other, and moreover the first differences are equal, then eight times the product of the least and the sum of all, added to the square of the least, is equal to the square of the sum of the least and twice the
The reference to Stevin is to his L'arithmétique ... aussi l'algebre (Stevin . Pages 558–642 contain Stevin's treatment of the 'Quatriesme livre d'algebre de Diophante d'Alexandrie'. On page 634, Stevin has the following Theorem.
Nombre triangulaire multiplié par 8, & plus 1 faict quarré a sa racine
A triangular number multiplied by 8, plus 1, makes a square commensurable with its
The reference to Maurolico is to his Arithmeticorum libri duo (Maurolico 1575, Prop .
Omnis triangulus octuplatus cum unitate, conficit sequentis imparis
Eight times any triangular number, plus one, makes the square of the next odd
As an example, Maurolico gave ; note that 15 is the fifth triangular number, 11 is the sixth odd number.
There is also a reference lower down to Maurolico's Proposition (Maurolico 1575, Prop :
Omnis numerus triangulo deinceps: aequatur quadratis lateris trianguli
Every triangular number joined with the preceding triangular number makes the square of its
Maurolico's example is . ]
The refrence to Viète is to his Variorum responsorum liber VII, Chapter IX, Proposition 14 (Viete 1593d, Chapter 9, Prop .
Propositio XIV.
Si fuerint lineae quotcunque æqualiter sese excedentes, fit autem prima excessui æqualis: octuplum ejus quod fit sub minima & composita ex omibus, adjunctum minimae quadrato, æquatur quadrato compositæ ex minima & extrema
If there are any number of lines exceeding each other, and moreover the first differences are equal, then eight times the product of the least and the sum of all, added to the square of the least, is equal to the square of the sum of the least and twice the
The reference to Stevin is to his L'arithmétique ... aussi l'algebre (Stevin . Pages 558–642 contain Stevin's treatment of the 'Quatriesme livre d'algebre de Diophante d'Alexandrie'. On page 634, Stevin has the following Theorem.
Nombre triangulaire multiplié par 8, & plus 1 faict quarré a sa racine
A triangular number multiplied by 8, plus 1, makes a square commensurable with its
The reference to Maurolico is to his Arithmeticorum libri duo (Maurolico 1575, Prop .
Omnis triangulus octuplatus cum unitate, conficit sequentis imparis
Eight times any triangular number, plus one, makes the square of the next odd
As an example, Maurolico gave ; note that 15 is the fifth triangular number, 11 is the sixth odd number.
There is also a reference lower down to Maurolico's Proposition (Maurolico 1575, Prop :
Omnis numerus triangulo deinceps: aequatur quadratis lateris trianguli
Every triangular number joined with the preceding triangular number makes the square of its
Maurolico's example is . ]
[Translation: ]
plutarchus platonica
quæstione 4a
[Translation: Plutarch on Plato, question ]
Vieta resp. pa.
[Translation: Viete, Liber variorum responsorum, page 15.
Maurolicus pr. 54.
[Translation: Maurolico, Arithmetica, Proposition 54.
Stevin . pag. 634. arith.
in
[Translation: Stevin, page 634, Arithmetic, on Diophantus.
quæstione 4a
[Translation: Plutarch on Plato, question ]
Vieta resp. pa.
[Translation: Viete, Liber variorum responsorum, page 15.
Maurolicus pr. 54.
[Translation: Maurolico, Arithmetica, Proposition 54.
Stevin . pag. 634. arith.
in
[Translation: Stevin, page 634, Arithmetic, on Diophantus.
Trianguli numeri octuplum, plus quadrato unitatis:
æqualie est quadrato
facto a duplo latere trianguli plus
[Translation: Eight times a trinagular number plus the square of one is equal to the square of double the side of the triangular number, plus one.
facto a duplo latere trianguli plus
[Translation: Eight times a trinagular number plus the square of one is equal to the square of double the side of the triangular number, plus one.
Sit latus trianguli
[Translation: Let the side of the triangle be .
Tum triangulus erit
[Translation: Then the triangular number will be
Unde propositio cum demonstratione in notis logisticis ita se
[Translation: Whence we have the proposition with its demonstration in arithmetic notation ]
[Translation: Let the side of the triangle be .
Tum triangulus erit
[Translation: Then the triangular number will be
Unde propositio cum demonstratione in notis logisticis ita se
[Translation: Whence we have the proposition with its demonstration in arithmetic notation ]
[Translation: ]
Marol. pr.
[Translation: Maurolico, Proposition ]
Omnis numerus triangulus, plus triangulo deinceps priori: æquatur
quadrato lateris trianguli
[Translation: Every triangular number, plus the triangular numebr following, is equal to the square of the side of the greater triangular number.
Sit latus maioris trianguli.
[Translation: Let the side of the greater triangular number be .
latus minoris triang:
[Translation: the side of the smaller triangular number is .
Ergo: maior triangulus.
[Translation: Therefore the greater triangular number is .
Minor triangulus.
[Translation: the smaller triangular number is .
Unde propositio cum demonstration in notis logisticis ita se
[Translation: Whence we have the proposition with its demonstration in arithmetic notation ]
[Translation: Maurolico, Proposition ]
Omnis numerus triangulus, plus triangulo deinceps priori: æquatur
quadrato lateris trianguli
[Translation: Every triangular number, plus the triangular numebr following, is equal to the square of the side of the greater triangular number.
Sit latus maioris trianguli.
[Translation: Let the side of the greater triangular number be .
latus minoris triang:
[Translation: the side of the smaller triangular number is .
Ergo: maior triangulus.
[Translation: Therefore the greater triangular number is .
Minor triangulus.
[Translation: the smaller triangular number is .
Unde propositio cum demonstration in notis logisticis ita se
[Translation: Whence we have the proposition with its demonstration in arithmetic notation ]
zoom in
zoom out
zoom area
full page
page width
set mark
remove mark
get reference
digilib