DUL. 
of tiie four sides. This proposition is of 
great use in the theory of compound moti- 
ons ; for, in an oblique angled parallelogram, 
the greater diagonal being tlie subtense of 
an obtuse, and the lesser of an acute angle, 
which is the complement of the former, if 
the obtuse angle be conceived to gi'ow till 
it be infinitely great with regard to the 
acute one, the great diagonal becomes the 
sum of the two sides, and the lesser one, 
nothing. Now two contiguous sides of a 
parallelogram being known, together with 
the angle they include, it is easy to find one 
of the diagonals in numbers, and tlien the 
foregoing proposition gives the other. This 
second diagonal is the line that w'ould be 
described by a body impelled at the same 
time by two forces which should have the 
same ratio to each other, as the contiguous 
sides have, and act in those two directions ; 
and the body would describe this diagonal 
in the same time, as it would have described 
either of the contiguous sides in, if only 
impelled by the force corresponding thereto. 
6. In any trapezium, the sum of the squares 
of the four sides is equal to the sum of the 
squares of the two diagonals together with 
four times the square of tlie distance be- 
tween the middle points of the diagonals. 
7. In any trapezium, the sum of the squares 
of the two diagonals is double the sum of 
the squares of two lines bisecting the two 
pairs of opposite sides. 8. In any quadri- 
lateral inscribed in a circle, the rectangle 
of the two diagonals is equal to the sum of 
the two rectangles under the two pairs of 
opposite sides. 
DIAL. Dials are of various construc- 
tions, some being horizontal, others vertical, 
and others moveable, so as to apply to any 
particular latitude at pleasure. The use 
of a dial is to indicate, the hour, which is 
done by means of a wire, or by a triangular 
board, &c. placed at right angles to the 
face or index. This triangular piece is 
called the stile, or gnomon, and is made to 
point due north : it should be perfectly ver- 
tical, and the dial’s face, on which the hours 
are marked, should be equally divided 
thereby; the line of 12 being in a true 
direction with the stile. This line of di- 
rection is called the substile : the angle 
contained between the summit of the stile, 
and the face of the dial is called the eleva- 
tion. All which have their planes, or faces 
paiallel with the horizon, are called hori- 
zontal dials; those which have perpendi- 
cular planes, or faces, are called vertical 
dials ; and such as are neither vertical, nor 
horizontal, are called reclining dials. When 
erect dials do not face either the north, or 
the south, they are called declinhig dials. 
An universal dial is one tliat answers for all 
latitudes. 
The line passing under the centre of the 
stile longitudinally, and marking the hour of 
12, is called the meridian : in declining dials 
the suhstile makes an angle witli the meri- 
dian, in proportion to the deviation from 
a iiortlierly direction : this angle is the dif- 
ference of longitude. With respect to the 
manner of constructing, and of placing 
these useful instruments, we shall now pro- 
ceed to give some account. 
The following is the most simple dial that 
can he made. (Fig. 1. Plate Dialling,) Di- 
vide a circle into twenty-four equal parts, 
and draw through tlie several points of divi- 
sion, rays from the centre. That point whicti 
is to be the north, is to be marked XII., 
the next on the right XI., and thus as far as 
V., or IIII. : those on the left of the south- 
ern point 12, are to be I., II., III., &c. 
in regular order down to VIII. In the 
centre, whence the circle was drawn, fix a 
pin, equal in length to about a diameter of 
tlie circle, and be very careful that it be 
perfectly upriglit. Now, placing the dial 
at such an elevation as may equal the latitude 
of the place where it is to be used, see 
that the Xllth hour be on the meridional 
line. Thus for the latitude of 50, the 
northerly part XII. would require to be 
raised 50 degi'ees from the horizon, so that 
the face of the dial would stand in the 
plane of the equator, and cause the shadow 
of the pin to fall on the index, thus to point 
out the time of day. This is called the 
equinoctial dial. 
The following is the best mechanical 
method known for making common hori- 
zontal dials. (Fig. 2.) Draw two concen- 
tric circles, between which the hours are 
to be marked, and assuming any point for 
the hour of XII., let the thickness of your 
gnomon, or stile, be set otF by two lines a b, 
c d passing near the centre e of the circle, 
perfectly parallel and equidistant from the 
meridional line, which passes exactly 
through the centre, and through the mid- 
day, or XII. point. Now cross the meri- 
dional line at right angles, by the VI. 
o’clock line /g, and from the points b and d, 
as centres, de.scribe the quadrants /A, gi, 
taking a good extent, so as to approacli 
nearly to the hour circle for the sake of 
minute division ; if such be required, divide 
the quadrants respectively into 90 equal 
parts; then from d draw a line at 11° f from 
the line c d, which will give the place of 
