Surd as a Power 8.3 86.
Would give negative, surd or imaginary values; Diophantus then traces how each element of the equation has arisen, and formulates the auxiliary problem of determining how the assumptions must be corrected so as to lead to an equation (in place of the "impossible" one) which can be solved rationally.
Thus the concrete fact required to enable us to pass arithmetically from the conception of a fractional number to the conception of a surd is the fact of performing calculations by means of logarithms.
Consideration of the binomial theorem for fractional index, or of the continued fraction representing a surd, or of theorems such as Wallis's theorem (ï¿½ 64), shows that a sequence, every term of which is rational, may have as its limit an irrational number, i.e.
On the one hand Aristotle by his doctrine of matter admitted a surd into his system.
On the other, he assigned to vas with its insight into rationality too high a function with regard to the concrete in which the surd was present, a power to certify the truth of scientific principles.
Since the fraction is infinite it cannot be commensurable and therefore its value is a quadratic surd number.
Conversely every positive quadratic surd number, when expressed as a simple continued fraction, will give rise to a recurring fraction.
Muir, The Expression of a Quadratic Surd as a Continued Fraction (Glasgow, 1874).
A number of this kind is called a surd; the surd which is the pth root of N is written ¦JN, but if the index is 2 it is usually omitted, so that the square root of N is written, /N.
This e is neither open nor close, but a surd e the pronunciation of which comes very near a.
In the same way the supporting vowel, which is regularly an e in CataIan, is often written a, especially after r (abra, ar bore m; astra, a s t r u m; para, p a t r e in); one may say that in the actual state of the language post-tonic e and a become indistinguishable in a surd sound intermediate between the French a and mute e.
Final d after a vowel has produced u (pea, p e d e in; niu, n i d u m; mou, to o d u m); buf when the d, in consequence of the disappearance of the preceding vowel, rests upon a consonant, it remains and passes into the corresponding surd; f r I g i d u s gives fred (pronounced fret).
B is replaced by the surd pat the end of a word (trobar in the infinitive, but trop in the present tense); so also in the interiOr of a word when it precedes a consonant (supvensr, s u b v e n i re, sopte, s u b t 0).
G guttural is replaced as a final letter by surd c (longa, but lone; trigar, but Inch).
One of the most notable differences between normal Portugiiese and Gahician is the substitution of the surd spirant in place of the sonant spirant for the Lat.
(2) Words terminating in s surd or sonant and in x anciently formed their plural by adding to the singular the syllable es (bras, brasses; pres,, preses; maleix, maleixes), but subsequently, from about the 15th centui-v, the Castilian influence substituted Os, so that one now hears brrissos, presos, ma~eixos.
The substitution of these interdental and guttural sounds for the surd and sonant spirants respectively did certainly not take place simultaneously, but the vacillations of the old orthography, and afterwards the decision of the Spanish Academy, which suppressed x (= I; x was retained for cs) and allows only c and g before e and i, I and j before a, a, a, make it impossible for us to follow, with the help of the written texts, the course of the transformation.
(v.) The further extension to fractional values (positive or negative) of n depends in the first instance on the establishment of a method of algebraical evolution which bears the same relation to arithmetical evolution (calculation of a surd) that algebraical division bears to arithmetical division.