## Alternate circles and their connection with the ellipse |

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15th Proposition absciss alternate circle alternate circle cut alternate circle STV angles Book of ELEMENTS Call centre chord circumference cut the axis Definition describe describe the circle diameter difference distance divided double draw drawn ellipse equal equation expressed external focus greater axis half internal joins the focuses less manner mean proportional meter method nate circle parallel parallelogram portion present principal produced proposed radius radius equal ratio rectangle contained reduced result right angles semi-dia semi-diameter shown sides simple equation simplicity solution square straight lines substitute tangent THEOREM.-If Third Book Treatise triangles Wherefore аа

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Page 49 - If two triangles have two sides of the one equal to two sides of the...

Page 1 - PROP. V. THEOR. The area of any circle is equal to the rectangle contained by the semidiameter, and a straight line equal to half the circumference. Let ABC be a circle of which the centre is D, and the diameter AC ; if in AC produced there be taken AH equal to half the circumference, the area of the circle is equal to the rectangle contained by DA and AH. Let AB be the side of any equilateral polygon inscribed...

Page 63 - ... a diameter, in a constant ratio, the author proves the following propositions relating to this curve : — 1. The rectangle of the abscissae is to the square of the ordinate, as the square of the semiaxis major to the difference of the squares of the semiaxis major and the excentricity. 2. The distance of any point in the curve from the focus, is to its distance from the directrix, as the excentricity is to the semiaxis major. 3. The sum of the distances of any point in the curve from the two...

Page 60 - H : and because E is the same multiple of A, that F is of B, and that magnitudes have the same ratio to one another which their equimultiples have ; (v.

Page 55 - ... bear to each other. But in the method now proposed, and resting upon the theory exhibited in the Treatise, the properties of the ellipse are to be expressed by signs and letters denoting the proportions that certain straight lines bear to each other. These will at first produce only simple equations.

Page 6 - ... and =5+2. When the number of terms is odd, the double of the middle term is equal to the sum of the two extremes, or any two terms equally distant from the middle term ; as in the former of the foregoing series 6x2=2 + 10, and =4+8.

Page 49 - ... equiangular, and have those angles equal which are opposite to the homologous sides.

Page 70 - The reactance x of any portion of a circuit is the square root of the difference between the squares of the...

Page 4 - ... initially coincide with a portion of the harmonic curve, well known for its utility in the resolution of a variety of problems of this kind. Now if the half length of the complete curve be called k, corresponding to a quadrant of the generating circle, and the greatest...