381191
[Commentary:
The demonstration on this page relies on Euclid, .
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 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. ]
De Continue proportionalibus. Et
[Translation: On continued propotions. An ]
[Translation: On continued propotions. An ]
Si fuerint quantitates continue proportionales; Erit ut terminus
rationis maior, ad terminum rationis minorem: ita differentia
compositæ ex omnibus et minimæ, ad differentiae compositæ ex
omnibus et
[Translation: If there are quantities in continued proportion: as the greater term of the ratio is to the lesser term of the ratio, so will be the difference between the sum and the least to the difference between the sum and the greatest.
rationis maior, ad terminum rationis minorem: ita differentia
compositæ ex omnibus et minimæ, ad differentiae compositæ ex
omnibus et
[Translation: If there are quantities in continued proportion: as the greater term of the ratio is to the lesser term of the ratio, so will be the difference between the sum and the least to the difference between the sum and the greatest.
Sint continue proportionales. . . . . . .
[Translation: Let there be continued proportionals , , , , , , .
Differentiæ compositæ ex omnibus
et
[Translation: The difference between the sum and the ]
Hoc est, omnes
[Translation: That is, all before ]
Differentæ compositæ ex omnibus
et
[Translation: The difference between the sum and the ] Hoc est, omnis
consequentes, quæ ita sub ante-
cedentibus
[Translation: That is, all the consequents, which are thus placed under the ]
[Translation: Let there be continued proportionals , , , , , , .
Differentiæ compositæ ex omnibus
et
[Translation: The difference between the sum and the ]
Hoc est, omnes
[Translation: That is, all before ]
Differentæ compositæ ex omnibus
et
[Translation: The difference between the sum and the ] Hoc est, omnis
consequentes, quæ ita sub ante-
cedentibus
[Translation: That is, all the consequents, which are thus placed under the ]
Inde manifesta Theorematis demonstratio; quia ut et ita (per
synthetica) omnes antecedentes ad omnes consequentes. (per 12. pr. li.
[Translation: Thus the demonstration of the theorem is clear; because as is to so, by construction, are all the antecedents to all the consequents (by Book 5, Proposition 12).
synthetica) omnes antecedentes ad omnes consequentes. (per 12. pr. li.
[Translation: Thus the demonstration of the theorem is clear; because as is to so, by construction, are all the antecedents to all the consequents (by Book 5, Proposition 12).
Datis . . , dabitur compositæ ex omnibus
quæ significetur per
[Translation: Given , , ..., , there is given the sum of all of them, denoted by .
quæ significetur per
[Translation: Given , , ..., , there is given the sum of all of them, denoted by .
Sit tota composita ducenda per primam
quantitatem affirmatam et secundum negatum:
facta erit . cæteræ partes
intermediæ factæ eliduntur quoniam æquales
affirmatæ et negatæ scilicet et
[Translation: Let the total be multiplied by the first quantity taken positively and the second taken negatively: it will make . The other intermediate terms will be destroyed because of equal positive and negative quantities like and .
quantitatem affirmatam et secundum negatum:
facta erit . cæteræ partes
intermediæ factæ eliduntur quoniam æquales
affirmatæ et negatæ scilicet et
[Translation: Let the total be multiplied by the first quantity taken positively and the second taken negatively: it will make . The other intermediate terms will be destroyed because of equal positive and negative quantities like and .
Ergo . compositæ ex omnibus. patet igitur
[Translation: Therefore is the sum of all of them. The demonstration is therefore clear.
[Translation: Therefore is the sum of all of them. The demonstration is therefore clear.
alia Notatio
[Translation: another notation for ]
[Translation: another notation for ]
Hinc in infinitis progressionibus cum hoc est ultima quantitatis in infinitum
abeat, tres isti termini sint proport[ionales] videlicet:
, , co[mpositæ om]nibus.
[Translation: Hence, in an infinite progression, since , that is, the last quantity, disappears to infinity, these three terms are clearly proportional: , , the sum of all.
abeat, tres isti termini sint proport[ionales] videlicet:
, , co[mpositæ om]nibus.
[Translation: Hence, in an infinite progression, since , that is, the last quantity, disappears to infinity, these three terms are clearly proportional: , , the sum of all.