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COR. (1). Hence AS. is a third proportional to AN, and the half of PN.
(Compare p. 377).
N.B.-If P,N, bounds the curve is sometimes called the base, and AN, the height of the curve. Also it is plain that if PN be produced, so that pNNP, then pN2: = 4 AS. AN, and therefore p is a point in the parabola; which is therefore symmetrical about AN1; the line Pp is called a double ordinate.
(2.) If ABCD (Fig. 72) be any rectangle, and if DC and CB are each divided into n equal parts, and let Dp, p of the equal parts into which CD is divided, and Bp2 =p of the parts into which CB is divided; join
Ap, meeting pag (parallel to AB) in P; then
P is a point in the parabola which has for
its base CB and height BA.
For, draw PN perpendicular to AB, and
Pin parallel to PN.
.. P is a point in the specified parabola by corollary to last article. The second of
the following practical methods is founded on the above principles. The first and third depend entirely on the definition of the parabola.
To describe a parabola, any absciss of the axis and the corresponding ordinate being given; or, the half width of the base (AD or DB), and the height of the curve (CD) being given.
1ST METHOD.-Bisect DB (Fig. 73) in the point E; join CE, and from E draw EF perpendicular to CE, and meeting CD produced
in F; make CO in DC produced and CF each equal to DF, F will be the focus of the required parabola. Take any number of points, 1, 2, 3, 4 (&c.), in CD; through them draw double ordinates, or lines perpendicular to the axis CD; then with the centre F, and radii OF, 01, 02, 03, 04 (0 &c.), describe arcs cutting the ordinates; the curve drawn through the points of intersection will be the parabola required.
2ND METHOD.-AB (Fig. 74) being the width at the base, and CD the height of the curve, as
before, construct the parallelogram ABEF, divide DA, DB and AF, BE respectively
into the same number of equal parts in the points
3 D 3 2 1
3RD METHOD.-Place a ruler GH (Fig. 75) at any convenient distance from C, parallel to the base AB, and take a piece of wood (called a square or set square), made in the form of a right-angled triangle IOK, placing the base IO against the ruler, and the other edge OK to coincide with the line CD; having found the focus F (as in the 1st method), fasten one end of a string at F, place a pencil at the point C, passing the string round it, and bringing the string back to K, fasten it to the end or point of the triangle; move the triangle or square along the ruler, keeping the pencil always against the edge of the square (as at E), and with the string stretched the pencil will describe
one-half of the curve. By reversing the square and proceeding in a similar manner, the other half may be drawn, and the parabola required completed.
If the reasoning employed in the case of the ellipse be carefully gone over, attending only to the difference that will result from the circumstance that e 7 1, the student will readily deduce the following expressions will be found to hold good of the hyperbola, which are entirely analogous to the corresponding expressions in the case of the ellipse.
It is on this (6th) property that the following practical construction is founded. The terms major axis, transverse axis, &c., in the hyperbola, are entirely analogous to the same terms in the ellipse.
To describe a hyperbola, the transverse (AB) and conjugate (CD) axes being given.
Through B (Fig. 77), at one end of the transverse axis AB, draw GH parallel to the conjugate axis CD, and make GH equal to CD; with the centre E and radius, equal to EH, describe a circle cutting AB, produced both ways, in the points F and ƒ, which will be the foci of opposite hyperbolas; take any number of points, 1, 2, 3, 4 (&c.), in AB produced, and with the centres F and ƒ, and radii B1, B2, B3, B4, (B &c.), and A1, A2, A3, A4 (A &c.), describe arcs cutting each other; the curve drawn through the points of intersection will be the hyperbola required.
The following consideration materially simplifies the construction of a hyperbola. Through A (Fig. 78) draw AD parallel and equal to BC, join CD and produce it indefinitely, take P any point in the curve, draw pPN,
meeting CD produced in p. Then
Now it is plain that PN and pN increase. as P is farther from A, or N from A, and therefore, the farther P is from the vertex the nearer it approaches CD produced. CD is called an asymptote-after a short distance the curve sensibly coincides with the asymptote. Hence, if in the practical construction above given EH be produced, it will serve as a guide to the curve, which can be drawn very accurately after a very few points have been determined by the construction.
Besides the conic sections, which we have briefly discussed above, there are several curves possessing curious or useful properties. Amongst the chief of these is the cycloid. The construction of this curve is sometimes useful to the artist. The following will suffice to explain the nature of the curve and the method of its construction.
The cycloid is a curve formed by a point in the circumference of a circle (called the generating circle), revolving on a straight or level line; it may be best described as the curve traced out by a point in the wheel of a carriage when in motion along a level road. When the generating circle revolves round another circle, the curve described by a point in the circle is then called an epicycloid, and is constructed in a similar manner to the cycloid.
To describe a cycloid, the diameter or width at the base (AB), and height (CD), of the
curve being given.
The most common method of describing the cycloid is by placing a ruler along the line AB, and taking a circle, such as a shilling, &c., according to the size or height of the curve required, and having fixed a point in that circle, to move it slowly along the ruler, marking different points in the curve, or by keeping a pencil fixed at the point chosen, and thus describing the curve. This, however, is liable to error, as the circle used very often slips, and cannot then revolve accurately; by construction, however, the curve may be correctly formed thus :
Let ACB (Fig. 79) represent the curve in question, AB its base, CD its height. Bisect CD in O, and through O draw EF parallel to AB; with centre O and radius OC describe a circle, and divide the circumference into any number of equal parts, as Cp1, P1 P2,... join Dp, Dp2... Again, divide AD into the same number of equal parts, DN, DN2 and draw O1N1 O2N2.
N2 Nz D
parallel to CD; with centre O, and radius CD describe a circle, and
with centre N, and radius Dp, describe an arc cutting the circle in the point P. This is a point in the curve, similarly with the centre 0, and radius CD. Another circle may be described and intersected by the arc of a circle described with centre N, and radius Dp1, and thus another point be determined; and so on for any number of points, which being joined carefully will give the curve in question.
We thus conclude our Treatise upon Practical Geometry. It has been our endeavour to confine it in extent to that which it is absolutely necessary the scientific draughtsman should be familiar with; and in the foregoing pages will be found all that is required for practical purposes,-no really essential propositions being omitted.