A Series on Elementary and Higher Geometry, Trigonometry, and Mensuration, Containing Many Valuable Discoveries and Impovements in Mathematical Science ...: Designed as a Text Book for Collegiate and Academic Instruction, and as a Practical Compendium on MensurationCollins brothers & Company, 1845 - Conic sections |
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Page 198
... conjugate diameter to the latter , and the two diameters are called a pair of conjugate diameters . 11. Any straight ... axis , and terminated both ways by the curve , is called an ordinate to the axis . 15. The segments into which the ...
... conjugate diameter to the latter , and the two diameters are called a pair of conjugate diameters . 11. Any straight ... axis , and terminated both ways by the curve , is called an ordinate to the axis . 15. The segments into which the ...
Page 203
... axis , major . That is , if Dd be a diameter conjugate to Pp , cutting SP in E , and HP in e , PE or Pe = Draw PF perpendicular to Dd , and HI parallel to Dd or Tt , cutting PF in O , AC . Then , since the angles at 0 are right angles ...
... axis , major . That is , if Dd be a diameter conjugate to Pp , cutting SP in E , and HP in e , PE or Pe = Draw PF perpendicular to Dd , and HI parallel to Dd or Tt , cutting PF in O , AC . Then , since the angles at 0 are right angles ...
Page 209
... axes . Let Pp , Dd , be any two conjugate diameters , ŠROX a parallelogram circumscrib- ing the ellipse formed by drawing tangents at P , D , P , R d ; then Pp , Dd , divide the par- allelograms SROX into four equal parallelograms . T ...
... axes . Let Pp , Dd , be any two conjugate diameters , ŠROX a parallelogram circumscrib- ing the ellipse formed by drawing tangents at P , D , P , R d ; then Pp , Dd , divide the par- allelograms SROX into four equal parallelograms . T ...
Page 210
... axis . That is , If Pp , Dd , be any two conjugate diameters , Pp * + ᎠᎴ = Aa + Bb . Draw PM , Dm , ordinates the axis . Then , by Cor . to Prop . XIV , AC + BC = CM2 + Cm2 + PM2 + Dm2 CP + CD .. 4AC ' + 4BC2 = 4CP + 4CD2 Or , Aa ' + ...
... axis . That is , If Pp , Dd , be any two conjugate diameters , Pp * + ᎠᎴ = Aa + Bb . Draw PM , Dm , ordinates the axis . Then , by Cor . to Prop . XIV , AC + BC = CM2 + Cm2 + PM2 + Dm2 CP + CD .. 4AC ' + 4BC2 = 4CP + 4CD2 Or , Aa ' + ...
Page 213
... axis and their abscissæ , PROPOSITION XXI . THEOREM . M E K The equal conjugate diameters of all ellipses described on the same axis major , all terminate in the same right lines . Let any number of ellipses HJ , FG be described on the same ...
... axis and their abscissæ , PROPOSITION XXI . THEOREM . M E K The equal conjugate diameters of all ellipses described on the same axis major , all terminate in the same right lines . Let any number of ellipses HJ , FG be described on the same ...
Other editions - View all
A Series on Elementary and Higher Geometry, Trigonometry, and Mensuration ... Nathan Scholfield No preview available - 2018 |
A Series on Elementary and Higher Geometry, Trigonometry, and Mensuration ... Nathan Scholfield No preview available - 2018 |
A Series on Elementary and Higher Geometry, Trigonometry, and Mensuration ... Nathan Scholfield No preview available - 2018 |
Common terms and phrases
abscissa altitude arithmetical progression axes base bisected chord circle circular circular segment circumference cone conjugate axis construction convex surface corresponding cosec cosine cylinder described diameter distance divided draw drawn ellipse equal to half equation expression feet find the solidity formed formula Geom geometrical given height hence hyperbola inches infinite series latus rectum length logarithm major axis middle frustum minor axis multiplied ordinate parabola paraboloid parallel parallelogram perpendicular plane portion prism PROBLEM Prop PROPOSITION pyramid quadrant quantity radii radius ratio rectangle revoloidal surface right angles Scholium sector segment sides similar similar triangles sine specific gravity sphere spherical triangle spheroid spindle square straight line tangent THEOREM tion transverse axis Trigonometry ungula versed sine vertex vertical virtual centre zone
Popular passages
Page 44 - Prove that parallelograms on the same base and between the same parallels are equal in area.
Page 197 - ... is such, that all straight lines drawn from a certain point within the figure to the circumference are equal to one another : 16. And this point is called the centre of the circle.
Page 81 - The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power.
Page 219 - To find the solidity of a hyperbolic conoid, or otherwise called a hyperboloid. RULE. To the square of the radius of the base, add the square of the diameter...
Page 68 - In the same way it may be proved that a : b : : sin. A : sin. B, and these two proportions may be written a : 6 : c : : sin. A : sin. B : sin. C. THEOREM III. t8. In any plane triangle, the sum of any two sides is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their difference. By Theorem II. we have a : b : : sin. A : sin. B.
Page 14 - ... this point of intersection, as a pole, and limited by the sides, produced if necessary. Let the angle BAC be formed by the two arcs AB, AC ; then it will be equal to the angle FAG formed by the tangents AF, AG, and be measured by the arc DE, described about A as a pole.
Page 27 - The circumference of every circle is supposed to be divided into 360 equal parts...
Page 36 - The solidity of a cylinder is equal to the area of its base multiplied by its altitude.
Page 7 - The radius of a sphere is a straight line, drawn from the centre to any point of the surface ; the diameter, or axis, is a line passing through this centre, and terminated on both sides by the surface.