FOR£ST AND STREAM. 
Yacht Designing.-XXXIIL 
BY W. P. STEPHENS. 
{Continued from page 494, yunc 24.; 
Of the many non-circular curves there are three whose 
names are familiar to all, and which are more or less use- 
ful in naval architecture. Each of the three is formed by 
the intersection of a plane surface with a cone whose 
base is a circle, and from this they are called the Conic 
Sections. 
The ellipse is practically a flattened circle, a complete 
and continuous curve, with what may be termed one long 
and one short diameter. Its distinguishing characteristic 
is that the sum of the distances of every point on the curve 
from two fixed points shall be equal to some given dis- 
tance. The two fixed points, E, F, Fig. 92, are termed the 
foci; the longer diameter, AB, passing through these 
points, is termed the major or transverse axis; the shorter 
diameter, CD, at right-angles to the longer, is termed the 
minor or conjugate axis. The sum of the distances of any 
point, G, from the two foci, GE and GF, must be equal to 
the sum of the .similar distances (as EH, FH) of any 
other point. The center of the ellipse is at the intersec- 
tion of the two axes, I. 
The parabola is a symmetrical curve of infinite extent, 
there being no limit to the length of the two extremities. 
Its characteristic is that every point on the curve shall be 
equally distant from some given point and a given straight 
line. In Fig. 93 the fixed point, F, is called the foc^is, the 
line, AB, is called the directrix, and any given point on 
the curve as C, must be equally distant from both F and 
AB ; that is, CD and CF must be equal. Any straight line 
perpendicular to the directrix and cutting the curve, as 
DC, is a diameter, and the diameter passing through the 
focus is termed the a.vis. 
The hyperbola, Fig. 94, is of infinite extent also; its 
characteristics are that the difference of the distances of 
any point from two given points shall be equal to a given 
distance. The two fixed points. A, B, are the foci; then if 
D and E represent any points on the curve, the line AF, 
or the difference between AD and BD, must be equal to 
AG, the difference of AE and BE. The point C, midway 
between the two foci, is the center. It is evident that 
from A and B two curves may be drawn, equal and 
opposite, so that for any two foci there is always a pair of 
twin hyperbolas. The line AB, passing through the foci, is 
the transverse axis, and any line, as DH, passing through 
the center and cutting both curves is a diameter. 
The hyperbola is very little used in naval architecture. 
The ellipse is the line of intersection of a plane with a 
cylinder as well as a cone, consequently the deck openings 
for raking masts or funnels and inclined rudder posts, or 
the hawse pipes through the bows, are all elliptical. It 
is also used for windows and portholes where space is 
limited in one direction, and for flat arches. 
The parabola is still more important in naval archi- 
tecture, from the fact that nearly all the curved of a 
vessel, the level and waterlines, and also the sections, are 
arcs of parabolas or of a parabolic nature; so that they 
may be treated as parabolas in describing them and com- 
puting their elements. 
While a true arc of a circle, described as in Problem 
XL, is frequently used for the curve of the sheer, an arc 
of a parabola is preferred by many as giving a more 
pleasing effect to the eye. Such a curve, as shown in B, 
Fig. 95, starts from a straight line and gradually increases 
in curvature to any desired degree. It gives a bold ap- 
pearance to the bow, and for a given height at the stem- 
head the rail is high forward of the mast without the ap- 
pearance of too much sheer. At the after end it curves up 
quickly in a graceful sweep that gives life to the counter. 
A circular sweep through the same points at bow, amid- 
ships and on the transom, is shown at A, Fig. 95, and also 
by the broken line in B. The waterlines of modern yachts 
are of this same parabolic form, comparatively straight 
forward and increasing to a sharp sweep in the run ; 
while the bow-buttock lines are of the same form, but re- 
versed, being full and round in the bow and running out 
very straight and clean in the after part of the under- 
water body, probably continuing as absolutely straight 
lines in the counter. The theoretic curve of a sail is also 
parabolic, with a. strong hollow next to the mast or stay, 
graduating into a flat sitrface about the leach. A section 
of a mainsail from mast to leach, taken along the line of 
the reefs or about parallel to the boom, should show a 
parabolic curve. 
Problem XV. — To describe an ellipse. The following 
are the best of numerous methods, the first mechanical, the 
second and third geometrical : 
First Method. — Lay off the major axis AB, Fig. 96; 
bisect it and lay off the minor axis CD. With half of AB 
as a radius, describe from D the two arcs cutting AB in 
F, F\ the foci. Drive a pin or small wire nail at each of 
the points F, F^, and make a loop of cord just long enough 
to pass around the two nails and to reach the point D. 
Place the point of a pencil within this cord at D, and move 
it steadily around the right and back to D again. It is 
customary to tie the ends of the cord to F and F\ but if 
this is done the pencil and cord must be shifted to draw 
the lower half of the ellipse. By using the continuous 
loop of cord the line maj' be drawn more smoothly and 
evenly and by one single stroke. This method will give a 
fair curve for the oval windows of a cabin house, or if the 
two nails are placed far apart and the loop made very 
short it will make the long, flat curve that is used on the 
transom of a rowboat. 
Second Method, Geometrical. — From the center E, Fig. 
97, with AE and DE as radii, describe the two circles 
touching the major and minor axis respectively. Draw a 
number of radii of the larger circle, not necessarily at 
equal distances. Let CE be one of these radii, then 
through C where it intersects the larger circle draw a line 
parallel to DE, and through F, the intersection with the 
smaller circle, draw a line parallel to AB ; the interse.ction 
of these lines will be a point of the ellipse and similar 
intersections for each of the other radii will give a suc- 
cession of such points. The radii should be spaced quite 
closely together near the ends of the major axis where the 
curve is most abrupt, but ^hey may be well apart liear 
the minor axis. 
Third Method, Geometrical. — ^Draw as befoire AB and 
CD, Fig. 98, and from D with the radius AE describe 
F/G. 92 
^ — ^ 
A 1 
' B — 
/ F/6.94 
F,c.95 
J \ 
the arms F and F\ giving the foci. From E, with the 
dividers set to any convenient distance, step off on'AE 
the equal intervals denoted by i, 2, 3, 4, First take Ai 
as a radius, and with the compasses describe two arcs, 
from F and F\ as centers. Take Bi as a radius and. 
describe similar arcs from the same centers and thus in 
turn with A2 and B2, etc. The various intersections of 
these arcs will give points on the required ellipse. 
In the various processes of yacht designing it is con- 
stantly necessary to ascertain the three different attributes 
of a surface, the area, center of gravity and moment, and 
the corresponding attributes of a solid, the cubic contents 
or volume, the center of gravity, and the m-oment. Thus, 
in designing a sail plan, as described in Part VII., it is 
necessary to ascertain the area of every sail, its center of 
gravity, and its moment about some assumed axis. In the 
same way, in dealing with a metal keel, it is necessary 
to ascertain the cubic contents, from which the weight 
may be calculated, the center of gravity, and the moment 
in relation 'to the hull proper. There are many different 
rules and processes for the computation of these elements 
in different surfaces and solids, but the following are the 
most useful in the ordinary practice of yacht designing : 
Areas of Figures. 
Parallelogram. — This* is the simplest of all geometrical 
figures. It may be a square, as Fig. 99, a, with four equal 
sides and four right angles; a rectangle, b, with opposite 
sides equal and adjoining sides unequal and four right 
angles ; a rhombus, c, with equal sides and acute and 
obtuse angles ; a rhomboid, d, with adjoining sides unequal. 
In all cases, as the name indicates, the opposite sides are 
parallel. The area is ascertained in the case of the square 
and rectangle by multiplying any two adjoining sides to- 
gether, the base by the altitude, or the length by the 
height. In the case of the two inclined figures, c and d, 
/ 
/ 
/ a 
FiG.dS 
the area is ascertained by multiplying the base, as before, 
by the least distance to its opposite side; the same as in 
the square and rectangle except that the height or altitude 
is now shorter than the side of the figure. 
Trapezoid.— Ixs. this figure only two sides are parallel, a-^ 
ir^ e. Fig. 99. The area is ascertained by adding together 
the two parallel sides, multiplying them by the least dis- 
tance between them, and dividing the product by 2. 
Triangle. — This figure has already been described in 
Part XXX. It will be evident that each of the figures a. 
b, c, d may be divided by a diagonal into two equal and 
symmetrical triangles, each of which is necessarily of one- 
half of the area of the original figure. The area of a 
triangle consequently is ascertained by multiplying any 
one side, the base, by the distance to the opposite angle, 
the perpendicular, or' altitude, and dividing the product 
by 2. This is true even in the case of an obtuse-angled 
triangle, Fig. 72, b, in which the altitude falls on the pro- 
longation of the base outside the figure. 
Trapezium. — This is an irregular figure, f, Fig. 99, with 
four sides, all unequal ; such as a gaff mainsail. The area 
is found by dividing it by a diagonal into two triangles 
and calculating the area of each by the preceding rule. 
Regular Polygon. — This figure, a pentagon, with five 
.sides; hexagon, with six; heptagon, with seven; octagon, 
with eight, etc., may have any number of sides, but all equal. 
Its area is ascertained by adding together all the sides, 
multiplying the sum by the perpendicular distance from 
the center to one side, and dividing by 2. This is, of 
course, equivalent to computing the area of one of the 
many equal triangles into which the figure may be divided 
and multiplying by the number of such triangles. 
Irregular Polygon. — This is a figure with any numlier 
of sides, Avhich are not all equal. Its area is calculated by 
dividing into triangles. 
Circle. — -The circumference of a circle is ascertained by 
multiplying the diameter by the figure 3.141591. The area 
of a circle may be ascertained with a very close degree of 
accuracy by multiplying the square of the diameter by 
the figure 0.7854. The square of a number is that number 
multiplied by itself. The area may also be ascertained by 
multiplying the square of the circumference by the figure 
0.07958. 
Ellipse.— Ih^ area of an ellipse may be ascertained very 
closely by multiplying the transverse and the conjugate 
diameters together and the product by the figure 0.7854. 
Parabola. — The area of a segment of a parabola may 
be ascertained with close approximation by multiplying 
the base by two-thirds of the altitude. 
Mr. N. L. Stebbins, the yacht photographer, recently 
made a short trip to England and Scotland, in the course 
of which he secured many specially interesting views a 
little outside the ordinary run of yacht phonography. 
Among these are portraits of Messrs. Watson and Thor- 
neycroft, each in his office; views of Mr. Watson's draft- 
ing room, and of portions of the Thorneycroft Works, 
and of other noted shipyards. 
