## Plane and Solid Geometry: To which is Added Plane and Spherical Trigonometry and Mensuration. Accompanied with All the Necessary Logarithmic and Trigonometric Tables |

### From inside the book

Results 6-10 of 48

Page 80

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**proportion**- al , and the homologous angles respectively equal . For , by the similarity of the triangles ABC and A'B'C ' , ACD and A'C'D ' , ADE and A'D'E ' , etc. , we deduce the following series of equal ra- tios : F E D A B E / D ... Page 83

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**proportion**, AB : AC :: BD ' : CD ' . For , drawing BE ' and CF ' perpendicular to AD ' , we have , by reason of ...**proportions**, AB : AC :: BD : CD , AB : AC :: BD ' : CD ' , just demonstrated , we deduce the following : BD : CD ... Page 84

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**proportion**, BC : AB :: AB : BD . The comparison of the triangles ABC , ACD will give , in like manner , BC : AC :: AC : DC . THEOREM XIV . In any right - angled triangle , the square , or second power , of the numerical value of the ... Page 85

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**proportion**, AB2 : AC2 :: BC × BD : BC × DC , or , :: BD : DC . In a similar manner , by combining the identical equation BC2 = BC2 , with the same two equations above , we have and BC2 : AB2 :: BC × BC : BC × BD , or , :: BC : BD ; X ... Page 90

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**proportion**is correct , we must have AFGD greater than AHKD ; on the contrary , it is less , hence the above**proportion**is impossible . Therefore , ABCD cannot be to AFGD as AB is to a line great- er than AF . By similar reasoning , we ...### Other editions - View all

### Common terms and phrases

a+b+c altitude apothem bisect centre chord circumference circumscribed cone consequently corresponding cosec Cosine Cotang cube cubic cylinder decimal denote diameter dicular divided draw drawn equation equivalent exterior angles feet figure frustum Geom give greater half hence hypotenuse inches intersection logarithm measure multiplied number of sides opposite parallel parallelogram parallelopipedon pendicular perimeter perpen perpendicular plane MN polyedral angle polyedron prism PROBLEM proportion pyramid quadrant radii radius ratio rectangle regular inscribed regular polygon respectively equal right angles right-angled triangle Scholium secant sector similar similar triangles Sine slant height solid solve the triangle sphere spherical triangle square straight line subtract suppose surface Tang tangent THEOREM three sides triangle ABC triangular prism volume ΙΟ

### Popular passages

Page 35 - 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 80 - 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. A D A' Hyp. In triangles ABC and A'B'C', To prove AABC A A'B'C' A'B' x A'C ' Proof. Draw the altitudes BD and B'D'.

Page 139 - If a straight line is perpendicular to each of two straight lines at their point of intersection, it is perpendicular to the plane of those lines.

Page 17 - The sum of all the angles of a polygon is equal to twice as many right angles as the polygon has sides, less two.

Page 176 - The radius of a sphere is a straight line, drawn from the centre to any point of the...

Page 182 - Every section of a sphere, made by a plane, is a circle.

Page 28 - If two triangles have two sides of the one equal respectively to two sides of the other, but the included angle of the first greater than the included angle of the second, then the third side of the first is greater than the third side of the second. Given A ABC and A'B'C ' with Proof STATEMENTS Apply A A'B'C ' to A ABC so that A'B

Page 165 - ... bases simply : hence two prisms of the same altitude are to each other as their bases. For a like reason, two prisms of the same base are to each other as their altitudes.

Page 29 - ... to two sides of the other, but the third side of the first greater than the third side of the second, the angle opposite the third side of the first is.

Page 13 - If equals be added to unequals, the wholes are unequal. V. If equals be taken from unequals, the remainders are unequal. VI. Things which are double of the same, are equal to one another.