721361
[Translation: ]
el. 5. pr:
[Translation: Elements, Book 5, Proposition 12.
el. 7. pr.
[Translation: Elements, Book 7, Proposition 12.
Si sint magnitudines quotcunque proportionales, Quemadmodum
se habuerit una antecedentium ad unam consequentium: Ita
se habebunt omnes antecedentes ad omnes
[Translation: If any number of magnitudes are proportional, then just as as one antecedent is to its consequent, so will the sum of the antecedents be to the sum of the ]
[Commentary:
On this folio, Harriot derives the sum of a finite geometric progression,
using Euclid
and its numerical counterpart, . He then extends his result to an infinite (decreasing) progression, by arguing that the final term must be infnitely small, that is, nothing.
If any number of magnitudes be proportional, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents.
If there be as many numbers as we please in proportion, then, as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents. ]
If any number of magnitudes be proportional, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents.
If there be as many numbers as we please in proportion, then, as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents. ]
1.) De progressione
[Translation: On geometric ]
[Translation: On geometric ]
[Translation: ]
el. 5. pr:
[Translation: Elements, Book 5, Proposition 12.
el. 7. pr.
[Translation: Elements, Book 7, Proposition 12.
Si sint magnitudines quotcunque proportionales, Quemadmodum
se habuerit una antecedentium ad unam consequentium: Ita
se habebunt omnes antecedentes ad omnes
[Translation: If any number of magnitudes are proportional, then just as as one antecedent is to its consequent, so will the sum of the antecedents be to the sum of the ]
Sint continue proportionales. , , , , ,
[Translation: Let the continued proportionals be , , , , , .
[Translation: Let the continued proportionals be , , , , , .
In notis universalibus
[Translation: In general notation we ]
. primum. p. primus terminus
[Translation: . first term. p. first term of the ratio.
. secunda. s.
[Translation: . second. s. second.
.
[Translation: . last.
.
[Translation: . all.
[Translation: In general notation we ]
. primum. p. primus terminus
[Translation: . first term. p. first term of the ratio.
. secunda. s.
[Translation: . second. s. second.
.
[Translation: . last.
.
[Translation: . all.
Ergo; si, p > s ut in progressi
[Translation: Therefore if p > s are in a decreasing progression:
[Translation: Therefore if p > s are in a decreasing progression:
Ergo; si, p < s ut in progressi
[Translation: Therefore if p > s are in an increasing progression:
[Translation: Therefore if p > s are in an increasing progression:
De infinitis progressionibus
decrescentibus in
[Translation: For a progression descreasing ]
Cum progressio decrescit et
numerus terminorum sit infinitus;
ultimus terminus est infinite
minimus hoc est nullius
[Translation: Since the progression decreases and the number of terms is infinite, the last term is infnitely small, that is, of no quantity.
[Translation: ]
decrescentibus in
[Translation: For a progression descreasing ]
Cum progressio decrescit et
numerus terminorum sit infinitus;
ultimus terminus est infinite
minimus hoc est nullius
[Translation: Since the progression decreases and the number of terms is infinite, the last term is infnitely small, that is, of no quantity.
[Translation: ]