Hints, Theoretical, Elucidatory, and Practical, for the Use of Teachers of Elementary Mathematics

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Whittaker & Company, 1840 - Euclid's Elements - 188 pages

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Page 75 - If two triangles have two sides of the one equal to two sides of the other, each to each, and the included...
Page 77 - Any two sides of a triangle are together greater than the third side. Let ABC be a triangle ; any two sides of it together are greater than the third side, viz., the sides BA, AC, greater than the side BC ; and AB, BC, greater than AC , and BC, CA, greater than AB. Produce BA to the point D, and make AD equal to AC ; and join DC.
Page 96 - Continue this process, till a remainder occur, which is contained exactly a certain number of times in the preceding one. Then this last remainder will be the common measure of the proposed lines ; and regarding it as unity, we shall easily find the values of the preceding remainders ; and at last, those of the two proposed lines, and hence their ratio in numbers. Suppose, for instance, we find GB to be contained exactly twice in FD ; BG will be the common measure of the two proposed lines. Put BG...
Page 115 - ... of terms that had embarrassed the greatest mathematicians, and would after a great number of revolutions, entirely change the figure of the Moon's orbit. From whence this important consequence is derived, that the Moon's mean motion and the greatest quantities of the several equations, will remain unchanged, unless disturbed by the intervention of some foreign or accidental cause These tracts are inscribed to the Karl of Maccleslield, President of the Royal Society.
Page 20 - ... in a course of infallible certainty and security. Each of these hasty glances must possess the clearness of intuitive evidence, and the certainty of mature reflection; and yet must leave the reasoner's mind entirely free to turn instantly to the next point of his progress. The faculty of performing such mental processes well and readily is of great value, and is in no way fostered by the study of logic.
Page 92 - The angles at the base of an isosceles triangle are equal to one another; and if the equal sides be produced, the angles -upon the other side of the base shall be equal.
Page 158 - By a line parallel to one of the sides of the triangle. Let ABC be the given triangle, to be divided into two parts, in the ratio of m to n, by a line parallel to the base AB. Make CE to EB as m to n : erect ED perpendicularly to CB, till it meet the semicircle described on CB, as a diameter, in B.
Page 87 - If from a point within a circle more than two equal straight lines can be drawn to the circumference, that point is the centre of the circle.
Page 23 - Four quantities are said to be proportionals, when the first is the same multiple, part, or parts, of the second, that the third is of the fourth.
Page 136 - Punctuality : by which is meant, that the performance of all exercises should be limited to a certain time, and then be rigorously exacted.

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