Elements of Geometry, and Plane and Spherical Trigonometry: With Numerous Practical Problems |
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Page iii
... divided into two ? We answer , that classifications and divisions are based upon differences , and that the differences seized upon for this purpose must be determined by the nature of the properties and relations we wish to investigate ...
... divided into two ? We answer , that classifications and divisions are based upon differences , and that the differences seized upon for this purpose must be determined by the nature of the properties and relations we wish to investigate ...
Page 33
... divided into two equal angles by the line CD ; then we have two A's , ADC and BDC , which have the two sides , AC and CD of the one , equal to the two sides , CB and CD of the other ; and C D = the included angle ACD , of the one , BOOK ...
... divided into two equal angles by the line CD ; then we have two A's , ADC and BDC , which have the two sides , AC and CD of the one , equal to the two sides , CB and CD of the other ; and C D = the included angle ACD , of the one , BOOK ...
Page 46
... divided into any number of parts , the rectangle contained by the two lines is equal to the sum of the several rectangles contained by the undivided line and the seve- ral parts of the divided line . D L K I H E F G B Let AB and AD be ...
... divided into any number of parts , the rectangle contained by the two lines is equal to the sum of the several rectangles contained by the undivided line and the seve- ral parts of the divided line . D L K I H E F G B Let AB and AD be ...
Page 47
... divided into any two parts , etc. This theorem may be proved algebraically , thus : Let w represent any whole right line divided into any two parts a and b ; then we shall have the equation w = a + b By squaring , w2 = a2 + b2 + 2ab ...
... divided into any two parts , etc. This theorem may be proved algebraically , thus : Let w represent any whole right line divided into any two parts a and b ; then we shall have the equation w = a + b By squaring , w2 = a2 + b2 + 2ab ...
Page 57
... CB2 = a2 + 2ax + x2 = a + x2 . But CD = a + x ; hence this equa- tion is equivalent to the equation AD × DB + CB2 = CD2 , which is the algebraic proof of the theorem . THEOREM XLV . If a straight line be divided into BOOK I. 57.
... CB2 = a2 + 2ax + x2 = a + x2 . But CD = a + x ; hence this equa- tion is equivalent to the equation AD × DB + CB2 = CD2 , which is the algebraic proof of the theorem . THEOREM XLV . If a straight line be divided into BOOK I. 57.
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Common terms and phrases
ABCD altitude angle opposite axis bisected chord circle circumference circumscribed common cone convex surface cos.a cos.c Cosine Cotang diagonal diameter dicular difference distance divided draw equal angles equation equiangular equivalent find the angles four magnitudes frustum given line greater half Hence the theorem homologous hypotenuse included angle inscribed intersect isosceles less Let ABC logarithm measured multiplied N.sine number of sides opposite angles parallelogram parallelopipedon pendicular perpen perpendicular plane ST polyedron PROB PROBLEM produced Prop proportion PROPOSITION prove pyramid quadrantal radii radius rectangle regular polygon right angles right-angled spherical triangle right-angled triangle SCHOLIUM secant segment similar sin.a sin.b sin.c sine solid angles sphere SPHERICAL TRIGONOMETRY square straight line Tang tangent three angles three sides triangle ABC triangular prisms Trigonometry vertex vertical angle volume
Popular passages
Page 322 - 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 30 - Therefore all the interior angles of the figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
Page 123 - In a given circle to inscribe a triangle equiangular to a given triangle. Let ABC be the given circle, and DEF the given triangle ; it is required to inscribe in the circle ABC a triangle equiangular to the triangle DEF.
Page 58 - If a straight line be divided into two equal parts, and also into two unequal parts; the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line.
Page 29 - If one side of a triangle is produced, the exterior angle is equal to the sum of the two interior and opposite, angles; and the three interior angles of every triangle are equal to two right angles.
Page 41 - 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 96 - The angle in a semicircle is a right angle ; the angle in a segment greater than a semicircle is less than a right angle ; and the angle in a segment less than a semicircle is greater than a right angle.
Page 65 - In a series of equal ratios, any antecedent is to its consequent, as the sum of all the antecedents is to the sum of all the consequents. Let a: 6 = c: d = e :/. Then, by Art.
Page 77 - FGL ; (vi. 6.) and therefore similar to it ; (vi. 4.) wherefore the angle ABE is equal to the angle FGL: and, because the polygons are similar, the whole angle ABC is equal to the whole angle FGH ; (vi.
Page 113 - From a given point, to draw a line parallel to a given line. Let A be the given point, and BC the given line.