# Algebra: An Elementary Text Book for the Higher Classes of Secondary Schools and for Colleges, Part 1

A. and C. Black, 1886 - Algebra
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Page 264 - To express that the ratio of A to B is equal to the ratio of C to D, we write the quantities thus : A : B : : C : D; and read, A is to B as C to D.
Page 324 - It follows that every rational integral equation of odd degree with real coefficients has at least one real root.
Page 140 - An improper fraction is one in which the degree of the numerator is greater than or equal to the degree of the denominator.
Page 463 - D; it can be demonstrated indirectly with the aid of 32. 34. It is required to find a point in a given straight line, such that the rectangle contained by its distances from two given points in the straight line may be equal to the rectangle contained by its distances from two other given points in the straight line.
Page 446 - Article ; for the product of the squares ot the differences of all the roots is made up of the product of the squares of the differences of the roots of...
Page 29 - If the quotient of two monomials be integral, its degree is the excess of the degree of the dividend over that of the divisor. For let A = C3?ymznup . . . A! = c'y?ym'zn'uf' . . . where c and c' are the coefficients, ;r, y,s,u.
Page 225 - The former denotes a family of straight lines whose distance from the origin is equal to a, the latter a circle whose centre is at the origin, and whose radius is equal to a. And here, as was noted generally by Lagrange, the singular solution seems to be, in relation to geometry, the more important of the two. 3. A more general class of problems is that in which it is required to determine the curves in which some one of the foregoing elements, Art. 1, is equal...
Page 234 - Moivre's theorem, (a + ib) (a' + ib') = /u/u' (cos (a + a') + � sin (a + a') } , which proves that the product of two complex] numbers is a complex number, whose modulus is the product of the two moduli, and whose amplitude is the sum of the two amplitudes.
Page 20 - The view which he takes of these laws is expressed by the phrase "canons of the science," as is evidenced by the following passage : ' ' As we have now completed the establishment of the fundamental laws of ordinary algebra, it may be well to insist once more upon the exact position which they hold in the science. To speak, as is sometimes done, of the proof of these laws in all their generality, is an abuse of terms. They are simply laid down as the canons of the science.
Page 318 - The product of a finite number of continuous functions is a continuous function so long as all factors remain finite.