Elements of Geometry |
From inside the book
Results 1-5 of 48
Page 86
... substitute the straight line PQ , we shall have the enclosing line APQB , less than APDQB . But , by hypothesis , this must be the shortest of all ; this hypothesis then cannot be maintained ; therefore each of the enclosing lines is ...
... substitute the straight line PQ , we shall have the enclosing line APQB , less than APDQB . But , by hypothesis , this must be the shortest of all ; this hypothesis then cannot be maintained ; therefore each of the enclosing lines is ...
Page 137
... substitute ABX AD , AO × AM , which will give : solid AG solid AZ :: AB × AD × AE : AO × AM × AX . Therefore any two rectangular parallelopipeds are to each other as the products of their bases by their altitudes , or as the pro- ducts ...
... substitute ABX AD , AO × AM , which will give : solid AG solid AZ :: AB × AD × AE : AO × AM × AX . Therefore any two rectangular parallelopipeds are to each other as the products of their bases by their altitudes , or as the pro- ducts ...
Page 150
... substitute five others , according to different combinations ; and there will result as many theorems , among which may be distinguished the following ; two triangular pyra- mids are similar , when they have their homologous sides ...
... substitute five others , according to different combinations ; and there will result as many theorems , among which may be distinguished the following ; two triangular pyra- mids are similar , when they have their homologous sides ...
Page 166
... substitute DB , and we shall have DB + DC or BC > AC . 2. If we suppose BC AC , we say that the angle BAC will be greater ABC . For , if BAC were equal to ABC , we should have BC = AC ; and if BAC were less than ABC , it would follow ...
... substitute DB , and we shall have DB + DC or BC > AC . 2. If we suppose BC AC , we say that the angle BAC will be greater ABC . For , if BAC were equal to ABC , we should have BC = AC ; and if BAC were less than ABC , it would follow ...
Page 177
... substitute the plane for the part taken away , and we shall have a new surface , which encloses the surface OABCD , and which would be less than PABCD . But this last is the least of all , by hypothesis ; consequently this hypothesis ...
... substitute the plane for the part taken away , and we shall have a new surface , which encloses the surface OABCD , and which would be less than PABCD . But this last is the least of all , by hypothesis ; consequently this hypothesis ...
Other editions - View all
Common terms and phrases
ABCD adjacent angles algebraic algebraic quantities altitude angle ACB base centre chord circ circle circular sector circumference coefficient common divisor cone consequently contains Corollary cube cylinder Demonstration denominator denoted diameter divided dividend division equal equivalent evident example exponent expression factors figure fraction frustum given gives greater greatest common divisor homologous sides inscribed less letters logarithm manner measure multiplied obtain parallel parallelogram parallelopiped perpendicular plane MN polyedron preceding prism proportion proposed equation proposition quotient radical sign radii radius ratio rectangle reduced regular polygon remainder result right angles Scholium side BC similar solid angle sphere spherical square root straight line substitute subtract suppose term THEOREM third tion triangle ABC triangular pyramids unity unknown quantity vertex whence
Popular passages
Page 63 - The square of the hypothenuse is equal to the sum of the squares of the other two sides ; as, 5033 402+302.
Page 7 - If two triangles have the three sides of the one equal to the three sides of the other, each to each, the triangles are congruent.
Page 151 - THE sphere is a solid terminated by a curve surface, all the points of which are equally distant from a point within, called the centre.
Page 76 - The perimeters of two regular polygons of the same number of sides, are to each other as their homologous sides, and their areas are to each other as the squares of those sides (Prop.
Page 25 - Two equal chords are equally distant from the centre ; and of two unequal chords, the less is at the greater distance from the centre.
Page 52 - The sum of the squares on the sides of a parallelogram is equal to the sum of the squares on the diagonals.
Page 160 - If two triangles have two sides and the inchtded angle of the one respectively equal to two sides and the included angle of the other, the two triangles are equal in all respects.
Page 203 - In every triangle the sum of the three angles is equal to two right angles.
Page 162 - In any spherical triangle, the greater side is opposite the greater angle ; and conversely, the greater angle is opposite the greater side.
Page 141 - If a pyramid is cut by a plane parallel to its base, the...