THEORY AND PRACTICE OF ENGINEERING.

Book II.

746

Fig.692.

with this difference, that one line falls on the concave andthe other on the convex: or, the two lines drawn fromthe foci to any point in the hyperbola, make equal angleswith the tangent on its opposite sides.

The Centre of a globe, AZlO, is that point which isequidistant from every part of its surface. The point Y isthe pole of the ’circle 4.

Lahire was the first who proposed the globular pro-jection, or the delineation of the terrestrial surface or anypart of it on a plane; it is important that this subjectshould be thoroughly understood. The projection of anycircle on the sphere, which does not pass through the eye,is a circle, and circles whose planes pass through the eyeare projected into straight lines. The angle made on thesurface of the sphere by two circles which cut each other,and the angle made by their projections, is equal.

Gnomonic or central projection is that where the eye issituated at the centre of the sphere, and the plane of pro-jection is a plane which touches the sphere at any pointassumed at pleasure : the point of contact is called theprincipal point, and the projections of all the other pointson the sphere are at the extremities of the tangents of thearcs intercepted between them and the principal point.

As the tangents increase very rapidly when the arcs ex-ceed 45°, and at 90° become infinite, the central projec-tion cannot be adopted for an entire hemisphere.

The Centre of a geometric square is 11, or the pointfrom which the greater circle is struck.

And the Centre of a rule S is in A, or the pivot onwhich it turns.

The Circumference is that line which bounds a circlewhose centre is A.

As we have seen there is a point in a circle, fromwhence a line may be drawn equidistant around it, andwhich is the circumference ; the rectification of the circle,or the determination of the ratio that the circumferencebears to the diameter cannot be expressed in finite numbers.

Archimedes in his treatise De Dimensione Circuli showedthat they were as 7 is to 22, or 113 to 355 : Dc Eagnayfound when the diameter was 1, that the circumferencewas 3-14159265358979323846264338327950288. Theareas of all circles are to one another in the ratio of thesquares of their diameters, or the area is one* fourth ofthe circumference : Archimedes makes it nearly in theproportion of 14 to 11 : the ratio the area of a circle bearsto the square of its diameter has been thus expressed,

2x4x4x6x6x8x &c.

3x3x5x5x7x7x &c.

8 24 48

which is the same as - x — x x &c ; the denominators

9 25 49

being the square of the odd numbers, and the numeratorsdiffering from the denominators by unity.

Circles have similar circumferences when their dia-meters or radii are equal, as in those of C D and F G, orV13 equal radius VE.

The greatest Circumference of a sphere is that whichis struck from the pole T as its centre, and which cutsit into two equal parts, the plane passing through thecentre H.

The curved surface of a Sphere is equal to the rectanglecontained by its versed sine, and the sphere’s circumference:for the fluxion of the surface is obviously equal to the rectangle contained by thefluxion of the circumference, and the circumference of the circle of which the radius is thesine ; it varies therefore as the sine ; but the fluxion of the cosine, or of the versed sine,varies as the sine, consequently the surface varies as the versed sine. Now where thetangent becomes parallel to the axis, the fluxion of the surface becomes equal to the rect-angle contained by the sphere’s circumference and the fluxion of the versed sine.

Fur. 693

rig. 694

Fig. 696.

Fig. 61)8.