Unknown-quantities sentence example

unknown-quantities
  • The device known as the method of least squares, for reducing numerous equations of condition to the number of unknown quantities to be determined, had been adopted as a practically convenient rule by Gauss and Legendre; but Laplace first treated it as a problem in probabilities, and proved by an intricate and difficult course of reasoning that it was also the most advantageous, the mean of the probabilities of error in the determination of the elements being thereby reduced to a minimum.
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  • He follows Vieta in assigning the vowels to the unknown quantities and the consonants to the knowns, but instead of using capitals, as with Vieta, he employed the small letters; equality he denoted by Recorde's symbol, and he introduced the signs > and < for greater than and less than.
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  • None of them, in point of fact, has held its ground, and even his proposal to denote unknown quantities by the vowels A, E, I, 0, u, Y - the consonants B, c, &c., being reserved for general known quantities - has not been taken up. In this denotation he followed, perhaps, some older contemporaries, as Ramus, who designated the points in geometrical figures by vowels, making use of consonants, R, S, T, &c., only when these were exhausted.
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  • On the other hand, Vieta was well skilled in most modern artifices, aiming at a simplification of equations by the substitution of new quantities having a certain connexion with the primitive unknown quantities.
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  • Thus, to investigate the composition of the system we must be able to calculate the value of r (n-1) unknown quantities.
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  • Now by elementary algebra we know that if the number of independent equations be equal to the number of unknown quantities all the unknown quantities can be determined, and can possess each one value only.
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  • Join to the original equations the new equation ax+(33'+yz=8; a like process shows that, the equations being satisfied, we have a,a,'Y,S a,b,c,d a' b ' c ' d'a",, b", c", or, as this may be written, a,13,y - 8a,b,c =07 a,b,c,d a',b'c' a' b r c r d'a", b N c" a", b", c", d",, , which, considering b as standing herein for its value ax+0y+yz, is a consequence of the original equations only: we have thus an expression for ax+/3y+yz, an arbitrary linear function of the unknown quantities x, y, z; and by comparing the coefficients of a, /3, y on the two sides respectively, we have the values of x, y, z; in fact, these quantities, each multiplied by a,b,c a' b ' c',, a N b r/ c", b", are in the first instance obtained in the forms a,b,c,d a 'b' c' d'a", b N' c N' dN,, , respectively.
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  • Assuming the mean motion of the moon to be known and the perigee to be fixed, three eclipses, observed in different points of the orbit, would give as many true longitudes of the moon, which longitudes could be employed to determine three unknown quantities - the mean longitude at a given epoch, the eccentricity, and the position of the perigee.
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