750

THEORY AND PRACTICE OF ENGINEERING.

Book

equation between the radius vector , which is a line drawnfrom the focus to the curve, and the angle which it makeswith the transverse axis: this is termed thepo/ur equationto the ellipse.

It is the property of this figure, that if a circle be de-scribed upon either axis, and from any point of that axisan ordinate be drawn, both to the circle and ellipse, thenthe ordinate of the circle is to the ordinate of the ellipse,as the axis to the other axis. Hence the whole area ofthe circle is to the whole area of the ellipse in the sameproportion, and consequently the area of an ellipse is amean proportional between the areas of the two circlesdescribed upon its transverse and conjugate axes.

The Oval G approaches nearer a circle than that shownat II, the transverse diameter N O being greater than thatofPQ.

An Ellipsis may be described by working a threadround the tw r o foci Y Z, and holding it at S, so that itmay pass TVXS, and thus form a true oval.

A Paraltola is a part of an oval cut off by a straightline, as 3hown in the figure 45(57, or at 123. Spirallines, as at S, bent in the manner of a volute, are struckfrom a variety of centres.

The parabola is also formed by the intersection of thecone with a plane parallel to one of its sides, and theterm is applied to all algebraic curves of a higher orderdetermined by an equation of the form y»"+»=?aw.r n .The curve whose equation is y :i = d i x is called the cubicalparabola, and that which has for its equation y -.= a.r 2 , thesemi-cubical parabola ; this latter curve was the first thatwas rectified or found equal in length to an assignablestraight line.

A figure is said to be inscribed within another when itis bounded, as that of the triangle D E F is bv the linesABC.

It is extremely useful at times to bound a regular aswell as an irregular figure within another whose dimen-sions arc known or can be easily computed: in anearly Italian edition of Vitruvius , we see the sectionsof the cathedral of Milan covered entirely with equilateraltriangles, for the purpose of accurately calculating itsquantity : our freemasons, particularly those who werenot thoroughly skilled in computation, could not adopta more simple means of ascertaining the area of a bodythan by applying an equilateral triangle to it; six suchwould form a hexagon, as the four in the cut do that ofthe triangle of similar sides ; this figure is capable of sub-division as well as multiplication, and presuming that thebase of a pillar was comprised within any such form, theproportion of its relative parts could be easily computed.Bounding an irregular figure with a parallelogram, andafterwards dividing it into a number of equilateral tri-angles, its area could be obtained, and with as much pre-cision as by numbers.

The Parallelogram LMNO has inscribed within it theirregular figure III K.

The Circle may have inscribed within it the square

PQSR.

In computing the relative areas of the two figures, wehave to consider only that the diameter of the circle isthe same as the diagonal of the square; to obtain whichwe have to square two of the sides and add them to-gether, and then extract the square root, which will bethe diameter of the circumscribing circle: when thesquare is formed on the outside of the circle, then theside of the square is the same as the diameter, and theirareas may be easily found by the ordinary rules.

Fig. 718.

Fig. 71*

Fig. 719.

v ''' T >-> . Q

Hv v

Fig. 721.