## Euclid and His Modern Rivals |

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### Common terms and phrases

adjacent angles 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 Pr Pr 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

### Popular passages

Page 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.

Page 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.

Page 34 - Thus, for" example, he to whom the geometrical proposition, that the angles of a triangle are together equal to two right angles...

Page 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.

Page 98 - ... angle. An acute angle is one which is less than a right angle.

Page 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.

Page 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...

Page 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.

Page 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.

Page 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...