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Elements of Algebra Preliminary to the Differential Calculus
Augustus De Morgan
No preview available - 2012
abbreviations algebraical quantities altered Anomaly answer applied arithmetical arithmetical meaning AUGUSTUS DE MORGAN called common Consequently contains convergent correct cube root cubic feet cubic foot denote differential calculus diminishes without limit Dionysius Lardner divergent series divided division divisor equal numbers example exponent functions given gives greater Hence impossible subtraction inches increases without limit infinite number instance Lemma less letters logarithm mantissa meaning method metical miles multiplied nearly equal negative quantity number of terms number or fraction numerator and denominator polynomial preceding equation problem proceed proved question rational rational polynomials remainder result rules shew shewn shillings a yard solution specific gravity square root stand student suppose supposition symbols taken tion true twelfth root unknown quantities Verification weight whence whole number word
Page xxviii - ... 2. From looking at the preceding instances, it appears that the rule of multiplication is as follows : Consider the first terms as having the sign + ; multiply every term of the multiplicand by every term of the multiplier, and put + before the products of terms which have the same sign, and — before the products of terms which have different signs. The preceding example is here written in the usual way, with the proper signs written to every term by the preceding rule. a + b — 2c d — a...
Page 64 - Then, f=|&, becomes, 5 : 7 : : 20 : 28, the numerator of one is, to its denominator, as the numerator of the other, to its denominator. It follows that any four numbers are proportionals, when the first divided by the second is equal to the third divided by the fourth. They are not proportionals when this is not the case ; therefore, by this test, we can determine if any four numbers are proportionals.
Page 251 - ... contained in this treatise, and those, for the most part, in so imperfect a way, that he is not fit to encounter any question unless he sees the head of the book under which it falls. On a very moderate computation of the time thus bestowed, the pupil would be in no respect worse off, though he spent five hours on every page of this work. The method of proceeding which I should recommend, would be as follows : Let the pupils be taught in classes, the master explaining the article as it stands...
Page v - The operation of division is also indicated by writing the divisor under the dividend with a line between them ; thus 14 by 2 is also frequently denoted thus y.
Page 228 - If the curiosity of any gentleman that has leisure, would prompt him to undertake to do the logarithms of all prime numbers under 100,000 to 25 or 30 figures, I dare assure him that the facility of this method will invite him thereto ; nor can anything more easy be desired. And to encourage him, I here give the logarithms of the first prime numbers under 20 to 60 places.
Page xxxix - ... arithmetic knows nothing, and therefore as we may suppose, uses language, finds methods, and adopts interpretations, of which arithmetic furnishes no examples. If the student have read* a little of geometry (a science which he * In England, the geometry studied is that of Euclid, and I hope it never will be any ether ; were it only for this reason, that so much has been written on Euclid, and all the difficulties of geometry have so uniformly been considered with reference to the form in which...
Page 2 - If equal numbers be added to equal numbers, the sums are equal numbers. That is, if a = b and c = d, then a + c = b + d, If a = b — c and x = p — g, then a + x = (b — c) + (p — q) = b +p — c — q.
Page 251 - ... failure of a first attempt to make the learner understand the principle .of a rule. It is no exaggeration to say, that under the present system, five years of a boy's life are partially spent in merely learning the rules contained in this treatise, and those, for the most part, in so imperfect a way, that he is not fit to encounter any question unless he sees the head of the book under which it falls. On a very moderate computation of the time thus bestowed, the pupil would be in no respect worse...