Euclid and His Modern RivalsMacmillan, 1885 - 275 sider |
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Side xiv
... proofs ; ( 3 ) New Propositions to be added . Proposed changes which , if proved to be essential , would necessitate such abandonment : - ( 1 ) Separation of Problems and Theorems ; ( 2 ) Different treatment of Parallels . 13 15 Other ...
... proofs ; ( 3 ) New Propositions to be added . Proposed changes which , if proved to be essential , would necessitate such abandonment : - ( 1 ) Separation of Problems and Theorems ; ( 2 ) Different treatment of Parallels . 13 15 Other ...
Side xix
... proof of Euclid's ( tacitly assumed ) Axiom , that two Lines cannot have a common segment Treatment of Angle دو · Parallels Assumption of R. Simpson's Axiom · Euclid's 12th Axiom replaced by a Definition , two Axioms , and five Theorems ...
... proof of Euclid's ( tacitly assumed ) Axiom , that two Lines cannot have a common segment Treatment of Angle دو · Parallels Assumption of R. Simpson's Axiom · Euclid's 12th Axiom replaced by a Definition , two Axioms , and five Theorems ...
Side xx
... proof of Playfair's Axiom discussed دو General survey of book : - Enormous amount of new matter Two non - sequiturs ' . An absurdity proved à la Henrici Motion ' per saltum ' denied A strange hypothesis . A new kind of ' open question ...
... proof of Playfair's Axiom discussed دو General survey of book : - Enormous amount of new matter Two non - sequiturs ' . An absurdity proved à la Henrici Motion ' per saltum ' denied A strange hypothesis . A new kind of ' open question ...
Side xxii
... proofs List of Euclid's Propositions which are omitted General survey of book : - A false Corollary A plethora of negatives A superfluous datum Cumbrous proof of Euc . I. 24 An unintelligible Corollary · A unique Theorem of equality ' A ...
... proofs List of Euclid's Propositions which are omitted General survey of book : - A false Corollary A plethora of negatives A superfluous datum Cumbrous proof of Euc . I. 24 An unintelligible Corollary · A unique Theorem of equality ' A ...
Side xxviii
... proof for Ax . 10 accepted . 217 . 218 • " ; § 6. Omissions , alterations , and additions , suggested by Modern Rivals . 39 219 Omission of I. 7 suggested . 220 Reasons for retaining it : - needed to prove I. xxviii ARGUMENT OF DRAMA .
... proof for Ax . 10 accepted . 217 . 218 • " ; § 6. Omissions , alterations , and additions , suggested by Modern Rivals . 39 219 Omission of I. 7 suggested . 220 Reasons for retaining it : - needed to prove I. xxviii ARGUMENT OF DRAMA .
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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...