723362
[Translation: ]
el. 9. pr:
[Translation: Elements Book IX, Proposition 35
Si sint quotlibet numeri deinceps proportionales, detrahuntur autem
de secundo et ultimo æquales ipsi primo: erit quemadmodum
secundi excessus ad primum, ita ultima excessus ad omnes qui ultimum
[Translation: If there are as many numbers as we please in proportion, and the first is subtracted from the second and the last, then just as the difference of the second is to the first, so is the difference of the last to all before the last.
[Commentary:
In the preceding folio, Add MS
f. , Harriot derived a formula for the sum of a finite geometric progression based on Euclid . Here he gives an alternative derivation based on Euclid .
If as many numbers as we please be in continued proportion, and there be subtracted from the second and the last numbers equal to the first, then as the excess of the second is to the first, so will the excess of the last be to all those before it. ]
If as many numbers as we please be in continued proportion, and there be subtracted from the second and the last numbers equal to the first, then as the excess of the second is to the first, so will the excess of the last be to all those before it. ]
2.) De progressione
[Translation: On geometric ]
[Translation: On geometric ]
[Translation: ]
el. 9. pr:
[Translation: Elements Book IX, Proposition 35
Si sint quotlibet numeri deinceps proportionales, detrahuntur autem
de secundo et ultimo æquales ipsi primo: erit quemadmodum
secundi excessus ad primum, ita ultima excessus ad omnes qui ultimum
[Translation: If there are as many numbers as we please in proportion, and the first is subtracted from the second and the last, then just as the difference of the second is to the first, so is the difference of the last to all before the last.
Progressio
[Translation: An increasing ]
[Translation: An increasing ]
In notis universalibus: sit , primus: , secundus: , ultimus: ,
[Translation: In general notation, let be the first term; the second term; the last term; the sum.
[Translation: In general notation, let be the first term; the second term; the last term; the sum.
Progressio
[Translation: A decreasing ]
[Translation: A decreasing ]
In notis universalis
[Translation: In general notation we ]
[Translation: In general notation we ]
Vel: in notis magis universalis.
sit , primus terminus rationis. , secundus.
, maxumus terminus progressionis
, minimus.
[Translation: Or, in more general notation, let be the first term of the ratio, the second, the greatest term of the progression, the least. Then:
sit , primus terminus rationis. , secundus.
, maxumus terminus progressionis
, minimus.
[Translation: Or, in more general notation, let be the first term of the ratio, the second, the greatest term of the progression, the least. Then: