16784
[Commentary:
The first reference in the heading is to Michael Stifel's Arithmetica integra
(Stifel , page 15. For Stifel, a diagonal number was obtained by multiplying the first two entries of a Pythagorean triple. The diagonal number corresponding to the triple (3, 4, 5), for example, is . Stifel also defined Pythagorean triples by the ratio of the two shorter sides, in this case . He was able to write out two lists, or orders, of triples, one with the shorter side odd (, , and so on), the other with the shorter side even (, , and so on). Stifel claimed that all possible triples were included in these two orders.
The second reference in the heading, possibly added a little later, is to Johannes Praetorius (Johann Problema, quod iubet ex quatuor rectis lineis datis quadrilaterum fieri, quod sit in circulo (Praetorius . On the final page, Praetorius discusses the problem of constructing cyclic quadrilaterals with rational sides.
Harriot sets out to disprove Stifel's claim, by demonstrating the existence of new orders of triples.
His first order (ordo. 1.) is the same as Stifel's first order. The triples are set out in three columns with differences calculated between rows. This allows Harriot to extrapolate forwards, but also backwards to a starting triple (1, 0, 1).
The second order (ordo. 2.) is the same as Stifel's second order. Again the triples are set out in three columns with differences calculated between rows. As for the first order this allows Harriot to extrapolate backwards to a starting triple (4, 3, 5). This is the first triple of the first order with the first two entries interchanged. Perhaps this gave Harriot the idea of interchanging other pairs. Thus he begins a third and new order (ordo. 3. novus) with (12, 5, 8), which is the second triple from the first order with the first two entries interchanged. This order immediately contains (20, 21, 29), which was not included in either of Stifel's orders. The fourth order begins with (15, 8, 17), which is the first triple from the second order with the first two entries interchanged. And so on. By the end of the page, Harriot has six orders, with differences in the left column of 2, 4, 8, 6, 10, respectively. This seems to suggest to him a more systematic method of displaying the orders, which he goes on to do on the next page.]
The second reference in the heading, possibly added a little later, is to Johannes Praetorius (Johann Problema, quod iubet ex quatuor rectis lineis datis quadrilaterum fieri, quod sit in circulo (Praetorius . On the final page, Praetorius discusses the problem of constructing cyclic quadrilaterals with rational sides.
Harriot sets out to disprove Stifel's claim, by demonstrating the existence of new orders of triples.
His first order (ordo. 1.) is the same as Stifel's first order. The triples are set out in three columns with differences calculated between rows. This allows Harriot to extrapolate forwards, but also backwards to a starting triple (1, 0, 1).
The second order (ordo. 2.) is the same as Stifel's second order. Again the triples are set out in three columns with differences calculated between rows. As for the first order this allows Harriot to extrapolate backwards to a starting triple (4, 3, 5). This is the first triple of the first order with the first two entries interchanged. Perhaps this gave Harriot the idea of interchanging other pairs. Thus he begins a third and new order (ordo. 3. novus) with (12, 5, 8), which is the second triple from the first order with the first two entries interchanged. This order immediately contains (20, 21, 29), which was not included in either of Stifel's orders. The fourth order begins with (15, 8, 17), which is the first triple from the second order with the first two entries interchanged. And so on. By the end of the page, Harriot has six orders, with differences in the left column of 2, 4, 8, 6, 10, respectively. This seems to suggest to him a more systematic method of displaying the orders, which he goes on to do on the next page.]
Examinatur Stifelius
de numeris diagonalibus. pa. 15
et prætorius. pag.
[Translation: An examination of Stifel on diagonal numbers, page 15, and Praetorius, last ]
de numeris diagonalibus. pa. 15
et prætorius. pag.
[Translation: An examination of Stifel on diagonal numbers, page 15, and Praetorius, last ]
ordo. 1.
[Translation: Order 1, ]
[Translation: Order 1, ]
ord. 2.
Platonic
[Translation: Order 2, ]
Platonic
[Translation: Order 2, ]
hoc
[Translation: that ]
[Translation: that ]
Dixit quod rationes omni
laterum sunt in istis
duobus
[Translation: He said that all the ratios of sides are in these two ]
Ego dico quod
[Translation: I say that is not ]
ordines sunt alij
Ut per
sequentia
[Translation: There are infinitely many other orders. As is clear from what ]
laterum sunt in istis
duobus
[Translation: He said that all the ratios of sides are in these two ]
Ego dico quod
[Translation: I say that is not ]
ordines sunt alij
Ut per
sequentia
[Translation: There are infinitely many other orders. As is clear from what ]
ordo. 1.
ordo. 2.
ordo. 3.
[Translation: Order 1.
Order 2.
Order 3, ]
ordo. 2.
ordo. 3.
[Translation: Order 1.
Order 2.
Order 3, ]
Melior est dispositio ordinum
in alijs chartis
[Translation: The arrangement of orders is better in the other sheets ]
in alijs chartis
[Translation: The arrangement of orders is better in the other sheets ]
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