NUMBERS. 
rable with unity ; as a number, incommen- 
surable with unity, is termed irrational or a 
surd. See Surd. In the samp manner a 
rational whole number is that whereof unity 
is an aliquot part ; a rational broken num- 
ber, that equal to some aliquot part of unity ; 
and a rational mixed number, that consist- 
ing of a whole number and a broken one. 
Even number, that which may be divided 
into two equal parts without any fraction, 
as 6, 12, Sec, The sum, difference, and pro- 
duct of any number of even numbers, is al- 
ways an even number. An evenly even 
number, is that which may be measured, or 
divided, witliout any remainder, by another 
even number, as 4 by 2. An unevenly even 
number, when a number may be equally 
divided by an uneven number, as 20 by 6. 
Uneven number, that which exceeds an 
even number, at least by unity, or which 
cannot be divided into two equal parts, as 
3, 5, Sec. The sura or difference of two un- 
even numbers make an even number ; but 
the factum of two uneven ones make an 
uneven number. If an even number be 
added to an uneven one, or if the one be 
subtracted from the other, in the former 
case the sum, in the latter the difference, is 
an uneven number; but the factum of an 
even and uneven number is even. The 
sum of any even number of uneven numbers 
is an even n\imber ; and the sum of any un- 
even number of uneven numbers is an un- 
even number. Primitive, or prime num- 
bers, are those only divisible by unity, as 
7, Sec. And prime numbers among them- 
selves, are those which have no common 
measure besides unity, as 12 and 1 9. Perfect 
number, that whose aliquot parts added to- 
gether make the whole number, as 6, 28 ; 
the aliquot parts of 6 being 3, 2, and 1,= 
6 ; and tliose of 28 being 14, 7, 4, 2, 1,= 
28. Imperfect numbers, those whose ali- 
quot parts, added together, make either 
more or less than the whole. And these are 
distinguished into abundant and defective ; 
an instance in the former case is 12, wdiose 
aliquot parts 6, 4, 3, 2, 1, make 16 ; and in 
the latter case 16, whose aliquot parts 8, 4, 
2, and 1 , make but 1.9. Plain number, that 
arising from the multiplication of two num- 
bers, as 6, which is the product of 3 by 2 ; 
and these numbers are called the sides of 
the plane. Square number, is the product 
of any number multiplied by itself : thus 4, 
which the factum of 2 by 2, is a square 
number. Every square number added to 
its root makes an even number. Polygonal, 
or polygonous numbers, the sums of arith- 
metical progressions beginning with unity : 
these, where the common difference is 1, 
are called triangular numbers ; where 2, 
square numbers; where 3, pentagonal num- 
bers ; where 4, hexagonal numbers ; where 
5, heptagonal numbers, &c. See Polygo- 
nal. Pyramidal numbers: the suras of 
polygonous numbers, collected after the 
same manner as the polygons themselves, 
and not gathered out of arithmetical pro- 
gressions, are called first pyramidal num- 
bers : the sums of the first pyraraidals are 
called second pyramidals, &c. If they arise 
out of triangular numbers, they are called 
triangular pyramidal numbers; if out of 
pentagons, first pentagonal pyramidals. 
From the manner of summing up polygonal 
numbers, it is easy to conceive how the 
prime pyramidal numbers are found, cii. 
(« — S)n^-f-Sn^ — (a — 5) re 
' — expresses all 
the prime pyramidals. 
Number of direction, in chronology, some 
one of the 35 numbers between the Easter 
limits, or between the earliest and latest 
day on which it can fall ; i. e. between the 
22d of March and the 25th of April. Thus, 
if Easter Sunday fall as in the first line be- 
low, the number of direction will be as on 
the lower line. 
March. April. 
Easter-day 22, 23, 24, 25, 26, 27, 28, 29, SO, 31. 1, 2, 3, &c. 
Number of direction 1, 2, 3, 4, 
and so on till the number of direction and 
the sura will be so many days in March for 
the Easter-day ; if the sum exceed 31, the 
excess will be the day of April. To find 
the number of direction: enter the follow- 
i, 6, 7, 8, 9, 10, 11, 12, 13, &c, 
ing table with the dominical letter on the 
left hand, and the golden number at top ; 
tlien where the columns meet is the nmnber 
of direction for that year. 
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