Euclid and His Modern RivalsMacmillan, 1885 - 275 sider |
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Side ix
... reasons for retaining , in all its main features , and specially in its sequence and numbering of Propositions and in its treat- ment of Parallels , the Manual of Euclid ; and thirdly , that no sufficient reasons have yet been shown for ...
... reasons for retaining , in all its main features , and specially in its sequence and numbering of Propositions and in its treat- ment of Parallels , the Manual of Euclid ; and thirdly , that no sufficient reasons have yet been shown for ...
Side x
... reason- ableness of this immemorial law : subjects there are , no doubt , which are in their essence too serious to admit of any lightness of treatment - but I cannot recognise Geo- metry as one of them . Nevertheless it will , I trust ...
... reason- ableness of this immemorial law : subjects there are , no doubt , which are in their essence too serious to admit of any lightness of treatment - but I cannot recognise Geo- metry as one of them . Nevertheless it will , I trust ...
Side xiv
... reasons for retaining Euclid's Manual . We require , in a Manual , a selection rather than a complete repertory of Geometrical truths Discussion limited to subject - matter of Euc . I , II . One fixed logical sequence essential One ...
... reasons for retaining Euclid's Manual . We require , in a Manual , a selection rather than a complete repertory of Geometrical truths Discussion limited to subject - matter of Euc . I , II . One fixed logical sequence essential One ...
Side xv
... Reasons assigned for separation Reasons for combination : - ( 1 ) Problems are also Theorems ; ( 2 ) Separation would necessitate a new numer- ation , ( 3 ) and hypothetical constructions . $ 4 . Syllabus of propositions relating to ...
... Reasons assigned for separation Reasons for combination : - ( 1 ) Problems are also Theorems ; ( 2 ) Separation would necessitate a new numer- ation , ( 3 ) and hypothetical constructions . $ 4 . Syllabus of propositions relating to ...
Side xvii
... Reasons for preferring Euclid's Axiom : — ( 1 ) Playfair's does not show which way the Lines will meet ; ( 2 ) Playfair's asserts more than Euclid's , the additional matter being superfluous . Objection to Euclid's Axiom ( that it is ...
... Reasons for preferring Euclid's Axiom : — ( 1 ) Playfair's does not show which way the Lines will meet ; ( 2 ) Playfair's asserts more than Euclid's , the additional matter being superfluous . Objection to Euclid's Axiom ( that it is ...
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Vanlige uttrykk og setninger
adjacent angles admit alternate angles angles are equal assert assume axiomatic beginners Cambridge Mathematical Tripos Certainly coincide coincidental Lines common point construct Contranominal course curve CUTHBERTSON deduce define Definition demonstration different directions different Lines draw drawn Elementary Geometry equal angles equally inclined equidistant equidistantial Euclid examining finite Lines given Line given point grant HENRICI infinite interior angles interpolated intersectional Lines Legendre less magnitude Manual mathematical mean meet if produced method Modern Rivals NIEMAND reads old proof omitted Pair of Lines parallel perpendicular Petitio Principii phrase Plane Plane Geometry Playfair's Axiom position Pref Problems Prop Propositions prove reductio ad absurdum remark right angles right Line separate point separational Lines sepcodal side straight angle straight Line suppose Syllabus text-book Th Th Theorem tion transversal Triangle true Wilson words writer دو وو
Populære avsnitt
Side 135 - If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz.
Side 143 - Your geometry states it as an axiom that a straight line is the shortest way from one point to another: and astronomy shows you that God has given motion only in curves.
Side 34 - Thus, for" example, he to whom the geometrical proposition, that the angles of a triangle are together equal to two right angles...
Side 201 - If there are three or more parallel straight lines, and the intercepts made by them on any straight line that cuts them are equal, then the corresponding intercepts on any other straight line that cuts them are also equal.
Side 98 - ... angle. An acute angle is one which is less than a right angle.
Side 203 - The sum of the squares on two sides of a triangle is double the sum of the squares on half the base and on the line joining the vertex to the middle point of the base.
Side 67 - Min. I accept all that. Nie. We then introduce Euclid's definition of ' Parallels. It is of course now obvious that parallel Lines are equidistant, and that equidistant Lines are parallel. Min. Certainly. Nie. We can now, with the help of Euc. I. 27, prove I. 29, and thence I. 32. Min. No doubt. We see, then, that you propose, as a substitute for Euclid's i2th Axiom, a new Definition, two new Axioms, and what virtually amounts to five new Theorems. In point of ' axiomaticity ' I do not think there...
Side 93 - Theorem. In every Triangle the greater side is opposite to the greater angle, and conversely, the greater angle is opposite to the greater side.
Side 203 - To construct a rectilineal Figure equal to a given rectilineal Figure and having the number of its sides one less than that of the given figure ; and thence to construct a Triangle equal to a given rectilineal Figure.
Side 200 - Assuming that the areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles...