5. In the marginal figure also, where cQ is a tangent to the parabola at the point c, and I K, O м, Q L, &c. parallel to the axis A D.

...

Then IEEK::CK: KL. (16) and a similar property obtains, whether CL be perpendicular or oblique to T D.

The external parts of the parallels I E, T A, O N, P L, &c. are always proportional to the squares of their intercepted parts of the tangent; that is,

the external parts I E, T A, 0 N, P L, are proportional to c 12, c T2, co', c P3, or to the squares C K2, C D2, C M2, C L3,

(17)

And as this property is common to every position of the tangent, if the lines I E, T A, O N, &c. be appended to the points I, T, 0, &c. of the tangent, and moveable about them, and of such lengths that their extremities E, A, N, &c. be in the curve of a parabola in any one position of the tangent; then making the tangent revolve about the point c, the extremities E, A, N, &c. will always form the curve of some parabola, in every position of the tangent.

The same properties, too, that have been shown of the axis, and its abscisses and ordinates, &c. are true of those of any other diameter.

6. PROB. To construct a Parabola.

D

A

B

с

Construct an isosceles triangle A B D, whose base A B shall be the same as that of the proposed parabola, and its altitude C D twice the altitude cv of the parabola. Divide each side A D, D B, into 10, 12, 16, or 20, equal parts [16 is a good number, because it can be obtained by continual bisections], and suppose them numbered 1, 2, 3, &c. from A to D, and 1, 2, 3, &c. from D to B. Then draw right lines 1, 1; 2, 2; 3, 3; 4, 4; &c. and their mutual intersection will beautifully approximate to the curve of the parabola A V B. Otherwise: by continued motion.-Let the ruler, or directrix в c, be laid upon a plane with the square G D O, in such manner that one of its sides D G lies along the edge of that ruler; and if the thread F M о equal in length to D o, the other side of the square have one end fixed in the extremity of the ruler at

Bi

X

D'

A

M

M' F

C

o, and the other end in some point F: then slide the side of the square D G along the ruler в c, and at the same time keep the thread continually tight by means of the pin м, with its part м o close to the side of the square D o; so shall the curve AM X, which the pin describes by this motion, be one part of a parabola.

And if the square be turned over, and moved on the other side of the fixed point F, the other part of the same parabola A M Z will be described.

7. PROB. Any right line being given in a parabola, to find the corresponding diameter: also, the axis, parameter, and focus.

Draw H I parallel to the given line D E. and G, through which draw A OG for the diameter.

Draw HR perp. to A G and bisect it in в; and draw v в parallel to a G, for the axis.

Make V BH в:: Hв: parameter to the axis.

Bisect D E, HI, in o

D

A V

F

E

G

R

H

B

Then the parameter set from v to F gives the focus. 8. PROB. To draw a Tangent to a Parabola. If the point of contact c be given, draw the ordinate c B, and produce the axis until a T = A B : then join T c, which will be the tangent.

Or if the point be given in the axis produced: take A B = A T, and draw the ordinate в c, which will give c the point of contact; to which draw the line T c as before.

If D be any other point, neither in the curve nor in the axis produced, through which the tan

gent is to pass: draw D E G perpendicular to the axis, and take D н а mean proportional between D E and DG, and draw нc parallel to the axis; so shall cbe the point of contact, through which and the given point D the tangent D C T is to be drawn.

DE H

T

B

G

When the tangent is to make a given angle with the ordinate at the point of contact: take the absciss A 1 equal to half the parameter, or to double the focal distance, and draw the ordinate IE also draw A H to make with A 1 the angle AHI equal to the given angle; then draw H c parallel to the axis, and it will cut the curve in c the point of contact, where a line drawn to make the given angle with c в will be the tangent required.

SECTION IV.-General Application to Architecture.

PROB. 1. To find, by construction, the position of the joints of the voussoirs, to a parabolic arch.

T

T

In the practice of arcuation, the voussoirs or arch-stones are so cut that their joints are perpendicular to the arch or to its tangent, at the points where they respectively fall. Hence, if A V в be the proposed parabola, P, P', P", &c. the points at which the positions of the joints are to be determined draw the ordinates P M, P' M', P" M", and on the prolongation of the axis. set off v TV M, V T'=V M', v т"=V м", &c. Join T P, T' P', T" P", &c. and perpendicular to them respectively the lines P o, p' o', p" o", &c. ; they will determine the positions of the joints required.

M

M

M"

PROB. 2. To find the same for an elliptical arch.

O"

Let A B be the span of the arch, and APP' P" B the arch itself, of which F and ƒ are the foci. Draw lines F P, fr, from the foci to each of the points P: bisect the respective angles FP f, FP'f, FP" f, by the lines P o, p' o', p" o"; they will show the positions of the joints at the points P, P', P".

Ꭺ F

PROB. 3. To find the same for a cycloidal arch.*

B

Let A V B be the cycloid, c p v q its generating circle, and P, P', P", points in the

arch where joints will fall. Draw the ordinates pm, p'm', r"m", each parallel to the base AB of the cycloid,

and cutting the circle

in the points p, p', p". Join v p, v p', v p'', and perpendicular to each the lines po, p' o', p" o"; parallel to each of which respectively draw Po, P'o', P" o"; they will mark the positions of the joints at the several points proposed.

* This problem is introduced here, as belonging to the subject of arcuation although it depends upon a property of the cycloid described hereafter, viz. that the tangent to any point P of a cycloid is parallel to the corresponding chord vp of the generating circle.

CHAPTER VI.

CURVES,

A knowledge of which is required by Architects and Engineers.

SECTION I.-The Conchoid.

Conchord, or Conchiles, is the name given to a curve by its inventor, Nicomedes, about 200 years before the Christian

era.

=

The conchoid is thus constructed: A P and B D being two lines intersecting at right angles: from P draw a number of other lines, P F DE, &c. on which take always DE DFA B or в c; so shall the curve line drawn through all the points E, E, E, be the first conchoid, or that of Nicomedes; and the curve drawn through all the other points, F, F, F, is called the second conchoid; though, in reality, they are both but parts of the same curve, having the same pole P, and four infinite legs, to which the line D B D is a common asymptote.

The inventor, Nicomedes, contrived an instrument for describing his conchoid by a mechanical motion: thus, in the ruler D D is a channel or groove cut, so that a smooth nail firmly fixed in the moveable ruler c A, in the point D, may slide freely within it into the ruler a P is fixed another nail at P, for the moveable ruler A P to slide upon. If therefore the ruler A P be so moved as that the nail D passes along the canal D D, the style, or point in A, will describe the first conchoid.

Conchoids of all possible varieties may also be constructed with great facility by Mr. Jopling's apparatus for curves, now well known.

Let A B=B C D E=D F=α, P B=b, B GE н=x, and G E =B H=y: then the equation to the first conchoid will be x2 (b+x)2 + x3 y2=a3 (b+x), or x+2 b x3 + b2 x2 + x3 y3 =a b2+2 a' bx+a x2; and, changing

only the sign of x, as being negative in the other curve, the equation to the 2d conchoid will be x2 (b. · x)2 + x3 y2 = a3 (bx), or x — 2 b x3 + b2 x2 + x3 y3 =ab2 2 a2 bx+a2 xo.

Of the whole conchoid, expressed by these two equations, or rather one equation only, with different signs, there are three cases or species: as first, when в c is less than BP, the conchoid will be as in the 2d fig. above; when B C is equal to в P, the conchoid will be as in the 3d fig.; and when B C is greater than в P, the conchoid will be as in the 4th or last fig.

Newton approves of the use of the conchoid for trisecting angles, or finding two mean proportionals, or for constructing other solid problems. But the principal modern use of this curve, and of the apparatus by which it is constructed, is to sketch the contour of the section that shall represent the diminution of columns in architecture.

The fixed point P is called the pole of the conchoid; D D D D the directrix: it is an asymptote to both the superior and the inferior conchoid. In the last figure the inferior conchoid is also nodated.

SECTION II.-The Cissoid or Cyssoid.

The cissoid is a curve invented by an ancient Greek geometer and engineer named Diocles, for the purpose of finding twe continued mean proportionals between two given lines. This curve admits of an easy mechanical construction; and is described very beautifully by means of Mr. Jopling's apparatus. At the extremity в of the diameter A B, of a given circle AO BO, erect the indefinite perpendicular e в E, and from the

Q