494 
FOREST ANO STREAM. 
[June 24, 1899. 
Yacht Designing^-XXXIL 
1^ BY W. P. STEPHENS, 
\Continued from page 463, ^une 10.; 
Problem VIII. — To describe an arc of given radius 
through two given points. Let A and B, Fig. 82, be the 
given points; set the compasses to the given radius, and 
from A and B in turn describe the short intersecting 
arcs, which will give the center of the circle. From this 
center with the same radius as before describe an arc 
which will cut the two points. 
Problem IX. — To find the center of a given circle. Let 
ABC, Fig. 83, be the circle ; draw any chord, as AB, bisect 
it and draw the perpendicular CD, the diameter; bisect 
this to find the center E. 
Problem X. — To find the center of an arc of a circlci 
Let ABC, Fig. 84, be the arc; from any two points, A and 
B, with a convenient radius, describe four intersecting 
arcs, and repeat the process with the opposite points B 
and C. Draw through these intersections the lines DE 
and GF, which will intersect at the center of the circle H. 
Problem XI. — To describe an arc of a circle through 
three points when the center is not accessible. It is fre- 
quently necessary to lay out a portion of a circle of such 
large diameter that it cannot be swept by anything in the 
nature of the compasses, beam compasses or trammels; 
as in the case of the deck beams of a large vessel. Several 
methods are used, the principal ones being as follows : 
First Method — Mechanical. — At two points of the re- 
quired circle, AB, some distance apart, pins or nails are 
driven, as in Fig. 85. Two straight rulers are then 
fastened firmly together at such an angle that, when each 
is in contact with one of the pins, the intersection of their 
outer edges, C, coincides with a third point of the re- 
quired arc. The point of a pencil is then held firmly at 
the intersection, and the rulers are moved, being always 
in contact with the pins, describing an arc of a circle. 
The apparatus may be made in permanent form, adjust- 
able to any circle, the two rulers being hinged at C with 
a brace and clamping screw to hold them in the required 
position. 
Second Method. — Let ADB, Fig. 86, be the required 
arc, as for instance Avhen the chord AB represents the 
length of a deck beam and the distance CD the round up 
of the beam. Draw BD, and at right angles to it BE; 
also DE parallel to AB. Divide the half chord CB into 
any number of equal parts, as at i, 2, 3 ; and DE into the 
same number of parts, i^, 2\ f. Join the corresponding 
points of division i, i\ etc. Draw BF perpendicular to 
AB, and divide it into the same number of equal parts as 
CB, as at a, b, c. Draw the lines Da, Db, Dc; the inter- 
section of I, and Da, of 2, 2^ and Db, and 3, 3^ and Dc, 
will be points of the required circular arc. 
Third Method.— In Fig. 87 draw DB, and from C the 
line CE perpendicular to DB; lay off FE equal to EC; 
then the angle EDB will be equal to the angle CDB 
Divide CD and DE into any number of equal parts. Draw 
BG perpendicular to AB and in length twice CD, and 
divide it into the same number of equal parts. The inter- 
sections of I, a, i\ B, etc., will be points of the curve. 
Fourth Method.— Let AB, Fig. 88, be the length, and CD 
the round. From C, with a radius SD, describe the arc 
D, a, b, c, E. Divide CE into any number of equal parts, 
as at I, 2, 3; and CB and the arc DE into the same 
number, at a, b, c. Also divide CB at i^, 2^ and 3\ and 
erect perpendiculars at each point. Take the distance 
I, a with the dividers and lay it off on i\ a^ and the 
other distances on their corresponding perpendiculars, 
2' , and 3 ' , ci' ; the points a' , b' , c' , will lie in the required 
curve. This method, and another similar one, but still 
less accurate, are only approximate; but the error is small 
when the span AB is great in comparison with the rise 
CD, as in the case of a deck beam ; and the operation is 
simpler and shorter than the preceding ones. 
After one-half of the required arc has been described by 
one of the above methods, the same operation may be 
repeated for the other half ; but the same end may be 
attained more speedily and easily by erecting perpendic- 
ulars at corresponding points of each half, as in Fig. 86, 
and transferring the distances already found in the first 
half to the proper positions in the second. 
Problem XII. — To describe a parabolic arc through two 
points. Let A and B, Fig. 89, be the points, and AC a 
tangent to the required curve. Draw CB and divide it 
into any number of equal parts, and divide AC into the 
same number. Erect perpendiculars at each point of 
division on AC, and join each point on CB with A by the 
lines A, i\ etc. Number the divisions of AC from A 
and those of CB from B, as shown; the intersections of 
the lines having similar numbers, i, i^, 2, 2\ etc., will be 
points of a parabolic curve tangent to AC at A. 
Problem XIII. — To describe a curve of versed sines. ' 
Let AB, Fig. 90, be the base of the required curve and 
AD its height. Bisect AD in C and from this center 
describe the semi-circle passing through A and D. Divide 
AB into any number of equal parts, and also the semi- 
circle AD into the same number of equal parts. The dis- 
tance A, a, the versed sine of the arc A, i^, is set off on 
the first perpendicular, i, a^; the distance A, h, the versed 
sine of A, 2\ is set off at 2, V, and each of the other dis- 
tances from AB to the points of division of the semi- 
circle is set off on its perpendicular, giving a series of 
points on the required curve. This curve is symmetrical, 
the. half BE being an exact reverse of DE, the two being 
tangent at E; while the curve is tangent at B to the 
base AB, and at D to a line parallel to AB. It is the 
easiest and most natural curve by which two parallel 
lines can be united. It is familiar by name to the tyro in 
designing in connection with the well-known investiga- 
tions and theories of the late J. Scott Russell and with 
the now accepted theory of the curve of areas that has re- 
placed them. 
Problem XIV. — To describe a trochoid. The trochoid 
is a fuller curve than the curve of versed sines, and while, 
in theory, the latter is used to govern the disposition of 
the displacement in the fore body of a vessel, the trochoid 
is used for the run or after body. Its construction is 
similar to that of the curve of versed sines, as in Problem 
XIII., uo to a certain point; but after the versed sine 
is set off, as at i, a^ Fig. 91, the sine of the arc a, i^, is 
set oft' horizontally, giving a point, a", on the required 
curve. In actual construction, after the arc AD and the 
base AB are divided, lines are drawn from A through 
the points of division of the arc A, i\ A, 2', etc., and a 
line parallel to each is drawn through the corresponding 
point of division; thus a line parallel and equal in length 
to A, 5^ is drawn through 5 and similarly through the 
other points. 
In dealing with any curve too large to be described by 
instruments, a number of points on the curve are located 
by means of measurements according to one of several 
methods. The simplest method is by means of what are 
termed rectangular co-ordinates, or abscissas and ordi- 
nates; the position of each point being determined by its 
distances from two given lines. Thus in Fig. 90 the 
point b^ may be located by measuring the distance A, 2 
along the line AB, and also the distance 2, b^ vertically 
from AB. The two lines AB and AD are known as the 
co-ordinate axes, the horizontal line AB being called the 
axis of abscissas, and the vertical line AD the axis of 
ordinates. The distance of a point from the axis of ordi- 
nates measured on a line parallel to the axis of abscissas, 
is called the abscissa of the point; and similarly, the dis- 
tance from the a.xis of abscissas, measured parallel to the 
axis of ordinates, is called the ordinate. Thus the abscissa 
of the point b^ is the distance b, b\ or its equivalent. A, 2; 
and the ordinate of the same point is 2, b' or A, b. 
The term ordinate is in constant use in designing, indi- 
cating the distance of a point from a line. Usually the 
line is horizontal, as in the sheer and half breadth plans, 
and the distances are measured vertically; but in some 
cases, as in the body plan, the ordinate may be the hori- 
zontal distance along a level line from the vertical center 
line. 
In the case of the curve DEB, Fig. 90, the abscissas 
would be the distances A, 5, A, 4, etc., along AB, and the 
ordinates the distances 5, d\ 4, c\ etc., measured on the 
perpendiculars through the different stations. 
In analytical geometry, from which the terms abscissa 
and ordinate are derived, the abscissas are measured from 
AD on or parallel to the line AB of indefinite length. In 
naval architecture the line AB is always assumed to be 
of a definite length, as for instance the load water line, 
and to be divided into an even number of equal parts. 
While the term ordinate is retained to indicate the dis- 
tance of a point from some given line, the term abscissa, 
which would practically indicate the distance along the 
line at which the measurement was made, is dropped. In 
place of it the term interval is used to indicate the length 
of the equal parts into which the line is divided, while 
the points marked by the intervals are called stations. In 
this way any curve may be readily measured, recorded 
and reproduced. A straight line is drawn in the general 
direction of the length of the curve. This base line is 
divided into a number of equal parts, perpendiculars are 
drawn to meet the curve, and the distances from the base 
line to each intersection are measured. 
A New Interclub Challeng^e Cup. 
The following circular has been sent out by the Eastern 
Y. C. : 
Boston, June 10.— To Members of the Eastern Y. C, 
the Regatta Committees of the New York, Seawanhaka 
Corinthian and Larchmont Yacht Clubs— Gentlemen : 
The Eastern Y. C, by its regatta committee, proposes to 
offer for competition a challenge cup substantially on the 
conditions stated below, and cordially asks you to send to 
us before June 30 any suggestions you may have to offer 
which would tend to produce better sport or a clearer un- 
derstanding of the terms. 
1. The Eastern Challenge cup is offered by the Eastern 
Y. C. as a perpetual challenge cup for competition by the 
New York, SeaAvanhaka Corinthian, Larchmont and East- 
ern yacht clubs, and such other American yacht clubs, 
having stations on salt water, as may from time to time be 
added to this list by the consent of the majority of the 
clubs already upon the list. 
2. M lie yachts in competition shall always be single- 
masted vessels, with centerboards or fixed keels, with a 
waterline length not less than 39 nor more than 46ft., with 
a racing measurement not exceeding 51ft., according to 
the rules of the club holding the cup at the time of 
challenge, and with fixed ballast (including its fasten- 
ings) not less than 20,ooolbs. 
3. No frames, beams or plating of such yachts shall 
be of any metal other than iron or steel. 
4. All challenges shall be in writing, and received be- 
fore Jan. I of the year of the proposed race by the club 
then holding the cup. 
5. Before April i the club holding the cup shall de- 
termine and announce the particular conditions of the 
races, which shall occur between Aug. i and Sept. 30. 
6. Each challenge shall be decided by winning a ma- 
jority in a series of three or five races, to be alternately 
triangular and windward and leeward, each leg to be not 
less than eight nautical miles in length. 
7. In case of several challenges for one year, the order 
of sailing the respective series may be determined by lot, 
and the winner of each series shall defend the cup against 
the challenger of the next appointed series; the winner of 
the final series to be the winner for the year. Each series 
in one year shall be at the place appointed by the club to 
which the challenge was sent, and under their rules and 
management. 
8. Before July 15 each challenging club shall notify the 
challenged club of the name and ownership of the yacht 
they have selected to represent them ; and within one 
week thereafter the challenged club shall announce the 
nanie and ownership of the yacht they have selected to de- 
fend the cup against each challenge. 
9. The parties to any challenge may agree upon any 
terms not inconsistent with paragraphs numbered 2 and 3 
of this deed of gift; and, excepting this paragraph and the 
paragraphs 2 and 3, the terms 6f the deed may be amended 
in any year between Oct. t and Dec. i by vote of three- 
fourths of the clubs then eligible to challenge. By vote, at 
any time, of all the clubs then eligible to challenge the en- 
tire deed may be changed. • 
10. If for any reason the club holding the cup at any 
time should be dissolved, the cup shall revert to the club 
which last previously held it. 
Please address any suggestions, before June 30, to the 
Regatta Committee, Eastern Y. C, Marblehead. Mass, 
Henry H. Buck, Chairman, 
Henry Howard, 
Odin B. Roberts, 
Eben B. Clarke, Secretary, 
' _ 95 Milk street. Boston, 
' ' ■ Regatta Committee, E. Y. C. 
East Gloucester Y. C. 
EAST GLOUCESTER, MASS. 
Saturday, June 17. • 
The East Gloucester Y. C. _ sailed a special race on 
June 17 in a light breeze, the times being: 
First Cls-Ss. 
Alethea, Colby & Tolman ." .....1 23 41 
Rambler 1 32 11 
Second Class. 
Snap Shot, Perry 1 35 04 
Tuton , ■. 1 59 05 
Third Class. 
Witch, Higgins 1 08 22 
Imp 1 08 39 
Baltimore Y. C. — Ramsay Cup. 
BALTIMORE — CHESAPEAKE BAY. 
Saturday, June 10. 
The Baltimore Y. C. sailed a race on June 10 for a cup 
presented by Vice-Com. Ramsay. The wind was fresh 
N.E., and the times were : 
Start. Finish. Elapsed. 
Albatross lO 58 48 1 11 3S 2 12 50 
Nepenthe .10 58 15 1 25 10 2 26 55 
Flossie 10 58 10 1 29 40 2 31 30 
Severn 11 01 00 1 45 50 2 44 50 
As the yachts had not been measured, the winner is not 
known. 
Cohasset Y. C* 
COHASSET. MASS. 
Saturday, June 17. 
The Cohasset Y. C. sailed its first race of the knock- 
about championship series on June 17 in a fresh S.E. 
breeze. The times were: 
Finish. 
Eleanor F, W. Moors 5 26 10 
Delta, R. G. Williams 5 26 30 
Nereid. W. R. Sears 5 27 35 
Baraceuta. Albert C. Burrage 5 30 01 
Remora, Crocker & Tower 5 31 20 
Hera,lda, A, Bigelow, Jr 5 32 24 
