?ro 
as 20, the product required. In lines it is 
alwa3's (and in numbers sometimes) called 
the rectangle between the two lines, or 
numbers, multiplied by one another. 
PROFILE, in architecture, the draught 
of a building, fortification, Scd wherein are 
expressed the several heights, widths, and 
thicknesses, such as they would appear, 
were thebuilding cut down perpendicularly 
from the roof to the foundation. 
Profile also denotes the outline of a 
figure, building, member of architecture, 
&c. Hence profiling sometimes denotes 
designing or describing the member with a 
rule, compass, &c. 
PRO 
Profile, in sculpture and painting, de- 
notes a head, portrait, &c. when represent- 
ed side ways, or in a side view. On almost 
all medals, faces are represented in profile. 
PROGNOvSTICS, among physicians, sig- 
nifies a judgment concerning the event of 
a disease, as whether it shall end in life 
or death, be short or long, mild or malig- 
nant, &Ci 
PROGRESSION, in mathematics, is 
either arithmetical or geometrical. Con- 
tinued arithmetic proportion, where the 
terras do increase and decrease by equal 
differences, is called arithmetic progres- 
sion. 
c a, „ + 6 „ 4- 2 6 a + 3 6, &c. increasing ) ^ difference d. 
)a, a — 6a — 2 0 0 — a 6, ike. decreasing^ ^ 
thus 
In numbers 
t 2, 4, 6, 8, 10, &c. increasing 7 j difference 2. 
i 10, 8, 6, 4, 2, &c. decreasing j ^ 
But since this progression is only a compound of two series, viz, 
, C Equals 
of I , 
u. 
I Arith. proportionals 0, i 6, ± 2 6, 4; : 
a, a, I 
&c. 
Ifl, 3, 5, t, 9, &c. a, a + b, a-}- 2 6, 
a _j_ 3 6, &c. a, a — 6, a — 2 6, a 3 6, 
&c, are in arithmetical progression. Hence 
it is manifest, that if a be the first term and 
u 6 the second, a -}- 2 6 is t he thir d, 
a -j- 3 6 the fourth, &c. and a ?i — 1 .6 
the a'** or last term. 
“ The sum of a series of quantities in 
arithmetical progression is found by multi- 
plying the sum of the first and last terms 
by half the number of terms.” 
Let a be the first term, 6 the common 
diflerence, n the number of terms, and s 
the sum of the series : Then, 
(I 4"U-|-'6 — ]— 0— j— 26 ...o-|- 7 i — 1.0 — Sf or, 
a-\-n — 1.6-|-a-t-u — 2.6-j-a-l-w — 3.6...-f-a=s. 
Sum, 2a-[-a — 1.6-{-2a-j-yi — 1.6-j-2a-j-?i — 1.6 
&c. to n terms, = 2s, 
or, 2 n-f-Ji — 1.6 XU — 2s. 
and s=:2a-l-K — 1.6 X- 
2 
Any three of the quantities s, o, n, 6, be- 
ing given, the fourth may be found from 
the equation sz=z2a-\-H — 1.6x 
Ex. 1. To find the sum of 18 terms of 
the series 1, 3, 5, 7, &c. 
Here a = 1, 6 = 2, « = 18 ; therefore, 
,s = 2-|- 34 X 9 = 324. ' 
Ex. 2. Required the sum of 9 terms of 
the series 11, 9, 7, 6, &c. 
In this case a = li, 6 = — 2, n = 9 j 
tlierefore s=5;2 — i6 X-== 6 x- = 27, 
2 2 
Ex. If thefirst term of an arithmetical pro- 
gression he 14, and the sum of 8 terras he 
28, wliat is the common difference ? 
Since 2a-\-n — 1.6 x 2 = 
— 1 ,6 X 
therefore, 6 = 
■ 1 .6 = — 2 - 
n 
-San 
$ s — S an 
2 s- 
In the case pro- 
n.n — 1 
posed, s = 28, a = 14, n : 
h — _^ 7 — SS __ 
■“8X7 ~ 7 — ~3. 
: 8 ; therefore. 
Hence, the series is 14, 11, 8, 5, &c. 
Progression geomctncal. Quantities are 
said to be in geometrical progression, or 
continual proportion, when the first is to 
the second, as the second to the third, and 
as the third to the fourth, Ax. that is, when 
every succeeding term is a certain multiple, 
or part of the preceding term. If a be the 
first terra, and ar the second, the series will 
be a, ar, ar^, ar\ ar\ &c. For « ; ar :: 
ur : ar ^ :: ar^ : ar^, &c. 
The constant multiplier is called the com- 
mon ratio, and it may be found by dividing 
the second term by the first. 
“ If quantities be in geometrical progres- 
sion, their differences are in geometrical 
progression." 
