Selected quad for the lemma: end_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	draw_v	equal_a	line_n	2,859	5	9.5591	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A65672	A new theory of the earth, from its original to the consummation of all things wherein the creation of the world in six days, the universal deluge, and the general conflagration, as laid down in the Holy Scriptures, are shewn to be perfectly agreeable to reason and philosophy : with a large introductory discourse concerning the genuine nature, stile, and extent of the Mosaick history of the creation / by William Whiston ...	Whiston, William, 1667-1752.	1696 (1696)	Wing W1696; ESTC R20397	280,059	488

be us'd about it 2. If in a considerably large Figure the two Points H and I be very near together it will be scarce distinguishable from a Circle and in any Figure if they be suppos'd to unite and be coincident the Eccentrical Curve will become Concentrical and the Ellipsis degenerate into a Circle as perfect a one as any drawn with a pair of Compasses Whence we see why a Circle is reckon'd among the Ellipses and how it may be generated by a way very like that made use of in their delineation 3. As when the Points H and I are coincident the Ellipsis loses its Eccentricity and denomination and commences a Circle so on the other hand if the distance H I be indefinitely lengthened while the difference between that distance and the length of the Cord equal to D H and I K or double to one D H as the Pencil at D is easily perceiv'd remains the same the Ellipsis will go through all Species and at last become indefinitely Oblong and Eccentrical and one half of it as F D E will degenerate into the very same Figure we call a Parabola For as all degrees of Eccentricity make Ellipses of all Species so no degree of Eccentricity makes a Circle and an indefinite or infinite degree of it makes a Parabola Which tho' we have no necessity to consider it so distinctly in this place none of the Heavenly Bodies as far as we yet know describing truly such a Line as has been already observ'd yet on account of the Comets Orbits which are nearly Parabolical at least deserv'd our notice and the first Figure will shew an example of it 4. An Ellipsis being describ'd about two Points as a Circle about one or those two united hence may appear in some measure the nature of these Points They are indeed called the Foci or Umbilici of the Figure but might not unfitly be nam'd the Centers thereof And how naturally each of them bears much the same respect to the Elliptick Periphery that the Center does to the circular one is partly obvious from the foregoing delineation and of which those who are acquainted with the Conick Sections cannot be ignorant To whom the matter will be still plainer if they consider the generation of an Ellipsis from the Section of a Conick superficies by a plain intersecting the opposite sides of the Cone and yet not parallel to the Basis as the Geometricians usually do For there the Axis of the Cone or Line which passes from its Vertex through the Center of the Circle its Basis does not pass through the middle or Center of the Ellipsis but one of those Points we are speaking of And accordingly if the name Center had not by custom in the Ellipsis been borrowed from the Circle on account of its position rather than some other properties of it and thence appli'd to the middle point in the Ellipsis it might very fitly as has been before said have been given to the two Points H and I now stil'd the Foci or Umbilici thereof And by the same reason the corresponding single Points going under the same names in the Parabola and Hyperbola would deserve and challenge the same denomination And this is so agreeable to the true System of the Planetary World that in the new Astronomy and thence in these Papers the stile is sometimes continued and 't is not unusual I may add nor very improper to say That the Sun the common Focus or Umbilicus of all the Celestial Elliptick Orbits is in the Center of our System or possesses the Center of the Planetary World 5. Tho' all the Lines passing through the Center in a Circle being equal are equally considerable yet 't is otherwise in the Conick Sections where that Line through the Focus alone which cuts the principal Axis at right Angles is remarkable above all the rest and in very many cases peculiarly considerable This Line is stil'd the Latus Rectum and in the Ellipsis is after the longer and shorter Axis the third proportional Thus in the Figure before us as DK is to EF so is the same EF to OP or MN the Latus Rectum thereof so famous with the Writers on the Conick Sections 6 The subtense of the Angle of Contact bd parallel to the distance from the Focus BH at an equal distance from the Point of contact B if that distance be suppos'd infinitely small is in all parts of the same Ellipsis or other Conick Section equal to it self The Truth and Use of which property is not yet sufficiently known 7. If from any Point in the circumference of an Ellipsis as B Lines be drawn to each Focus BH BI these two Lines taken together are always equal to themselves and to the longer Axis KD As the delineation of the Figure does plainly manifest 8. If the Angle made by the Lines to the Foci from any certain Point HBI be divided in the midst by the Line BA the said Line BA will be perpendicular to the Tangent or Curve at the Point of contact and so the Angles ABL ABG will be right ones and equal to each other as consequently will equal parts of them LBH IBG 9. A Line drawn from either Focus to the end of the lesser Axis HE or IE is equal to half the longer Axis CD or CK as is evident by the last particular but one And the same Line is Arithmetically the middle proportional between the greatest and least distance from the said Focus Thus HE for instance is just so much longer than HD as 't is shorter than HK the difference in both cases being the Eccentricity HC or CI. 10. The Tangent of an Ellipsis LG is never perpendicular to a Line drawn from the Focus excepting the two points which terminate the longer Axis D and K. And if you imagine the point of contact B with the Radius BH and the Tangent LG to move round the Ellipsis together from B towards D the preceding Angle HBL will in the descent from K by F to D be an acute one its acuteness increasing from K to F and as much decreasing from F to D and in the ascent from D by E to K an obtuse one its obtuseness increasing from D to E and as much decreasing from E to K in both semirevolutions arriving at rightness at the Points D and K the ends of the longer Axis alone as was here to be observ'd 11. The Area of an Ellipsis is to that of a circumscrib'd Circle whose Diameter is equal to the others longer Axis as the shorter Axis of the Ellipsis is to the same longer Axis or Diameter 12. If the Circumferences of a Circle and of an Ellipsis be equal the Area of the Circle is the greater It being known that of all Figures whose Perimeters are equal the Circle is the most capacious 13. If an Ellipsis by becoming infinitely Eccentrical degenerate into a Parabola the Latus Rectum will be