A reply to a critical and monthly reviewer, in which is inserted Euler's demonstration of the binomial theorem / by Abram Robertson.

  • Robertson, Abram, 1751-1826.
Date:
1808
Catalogue details

Licence: Public Domain Mark

Credit: A reply to a critical and monthly reviewer, in which is inserted Euler's demonstration of the binomial theorem / by Abram Robertson. Source: Wellcome Collection.

Provider: This material has been provided by Royal College of Physicians, London. The original may be consulted at Royal College of Physicians, London.

    13/42 (page 11)
    [—T = [;] quia autem i eft numerus integer, erit [z] = (i + .r); (vide §.4.) ficque erit [—]* = (1 + x)'1 unde ra- i ? dicem quadratam extrahendo fit [—] = (1 + x)~7 ficque jam tantum fumus confecuti, ut theorema Neutonianum etiam verum fit cafibus, quibus exponens n eft hujufmodi fradtio —. §• 10. Simili modo 'li ponamus 3m = i ut lit m = —, altera formularum fuperiorum prsebet [—]3 = [z] = 3 (* + x) hinc radicem extrahendo nancifcimur [—1 = 3 % (! + ■*) — ficque theorema noftrum etiam verum eft fi 3 exponens n fuerit hujufmodi fradtio —, atque hinc in ge- 3 nere manifeftum fore [—] = (i + x)~ ita ut jam demon- Cl ftratum lit, theorema noftrum verum efle, fi pro exponen- te n fradtio qusecunque — accipiatur, unde veritas jam eft evidta pro omnibus numeris pofitivis loco exponentis n accipiendis. §. ii. Supereft igitur tantum, ut veritas quoque often - datur pro cafibus, quibus exponens n eft numerus negati- vus. Hunc in finem in fubfidium vocemus redudtionem primo inventam [zzz] . [zz] = [zzz -f- n\ ubi denotet zzz, nu- merum pofitivum five integrum five fradtum ita ut fit uti modo oftendimus [m] = (i 4- x)«, deinde vero ftatuatur n = — m eritque m -f n = o ideoque [o] = (i -f- x)0 — jy quibus fubftitutis formula fuperior fuppeditat (i -f- x)m. [ — zzz] = i unde collierimus f — m~\ = I L J (i + x)’» ~ (I + x)-™ ficque etiam demonftratum eft theorema Neu- tonianum verum quoque effc, fi exponens n fuerit nume- rus