Monographs on Topics of Modern Mathematics: Relevant to the Elementary Field

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Jacob William Albert Young
Longmans, Green, and Company, 1911 - 416 Seiten
 

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Seite 87 - That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
Seite 178 - b, it is always possible to find some multiple of a which is " t/reater than " b. This fact is of great importance in the theory of measurement. 28. Classification of real points. Among the real points the points 1, 2, 3, ... [sec. 25, (15)] are called the positive integral points, and the points —1, —2, —3, . . . the negative integral points; all these, together with the point 0, form the subclass of " all integral points." All real points which can be expressed in the form ±m/n, where m...
Seite 35 - If one side of a triangle be produced, the exterior angle is greater than either of the interior, and opposite angles.
Seite 392 - ten decimals are sufficient to give the circumference of the earth to the fraction of an inch, and thirty decimals would give the circumference of the whole visible universe to a quantity imperceptible with the most powerful microscope.
Seite 356 - G(x) must be the product of <£(z) and a constant. The theorem in sec. 8 is therefore equivalent to the following: The unique equation, irreducible in the domain D, which is satisfied by a function Xi derived from numbers of D by a finite number of rational operations and extractions of square roots is of degree a power of 2. 11. From the last theorem and the criterion hi sec.
Seite 105 - B', B, or B" according as KD is less than, equal to, \ AB FIG. 19. .AL B' B B" FIG. IS. or greater than LB. In these three cases 2±D is greater than, equal to, or less than, 2±B respectively. Hence: Each summit angle of an isosceles birectangular quadrilateral is less than, equal to, or greater than, a right angle, according as the summit of the quadrilateral is greater than, equal to, or less than, the base. 18. In the Euclidean geometry each summit angle of an isosceles birectangular quadrilateral...
Seite 275 - Their graphs can be readily drawn without the use of a table, if it is noticed that y=a* can be plotted very easily, and that the graph of the logarithm can then be found by simply rotating the plane about a line through the origin making an angle of 45° with the x-axis. After these preliminaries the usual course in trigonometry "can be given with considerable economy in time on account of the familiarity which the student has already gained with graphical methods and the use of the logarithmic...
Seite 189 - C, © is defined as the ordinary <. This system satisfies all the postulates except Postulate 27. It is larger than the system of ordinary complex quantities, and contains that system just as the system of ordinary complex quantities contains the system of real quantities. Postulate 27 is therefore a restrictive condition. 36. What is algebra? We are now in a position to answer the question, " What is the algebra of complex quantities?
Seite 348 - Criterion. A proposed construction is possible by ruler and compasses if, and only if, the numbers which define analytically the desired geometric elements can be derived from those defining the given elements by a finite number of rational operations and extractions of real square roots.
Seite 26 - ... two triangles are congruent. The following theorems are proved similarly and are left as an exercise for the reader. Theorem 32.* // two sides and the included angle of one triangle are congruent to two sides and the included angle of anotlier triangle the two triangles are congruent.

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