NUMBERS. 
4. And, generally, as many different ways as there are of 
forming the sumber n by the addition of the numbers 1, 2, Fe Ay 
3 fe paced different ways may the number n + m be dt 
vided int m 
ways it may be di 3; th 
two queftions will be refolved, from the foregoing theorems, 
and thofe deduced from Prop. XXII. if we can afcertain in 
how many ways a harar pe be formed by the addition 
of the number » 3» c. m, from a comparifon of 
which we eafily deduce the "following 
THEOREMS. 
5. The number n may be parted into m unequal parts, in as 
side aa — may be formed by 
many ways as the number n — 
sae a the numbers ha 9 39 4) 
ber n may be parted into m uk or unequal 
par ey as ie ways as the number n — m m4 be formed by 
the addition of the numbers 1, 2 
rom thefe two theorems (which : até,” in ey th 
Janis theorems deduced from Props. XXII ixxiv, 
cept being in another form) we may ae tthefe other 
THEOREMS. 
7. The number n may be divided into m unequal parts, in as 
m 
many ways as the number n — 
m— I rae ‘ 
“a may be divided into m 
i equal, or unequal. 
. The number n may be divided into parts egual, or unequal 
in as many ways as the number n a 
~m— 
7 —— may be parted * 
into m unequal parts. Euler’s Analyfis Infinitorum. 
We fhall now conclude this article by enumerating a few 
curious properties of numbers, that could not properly be 
introduced under any of the preceding heads; but for their 
aie aio and demonftrations we muft refer the reader 
to works above quoted. 
Mifcellaneous Numerical alee 
. If a be a prime number, and «x mber whatever 
not divifible by a, then will «* ~* be ae "divifible by a. 
integer ae ape is ai fum of one, two, or three 
bers; of o 
him without demonftration, at page 180 of h 
Diophantus; the cafe of fquares has been proved both 
ee and La Grange, and that of the triangular numbers 
by Le Gendre ; beyond which it ftill remains without demon 
ftration, having re 
ara analy tts. 
3. If nbea - _ the ile 
+3 
refifted the efforts of feveral very diftin- 
eosese 
“3 
is divifible by Ny “fe aif is the produé 
ee ae ee ee CS -) 
2 
the subignons en being ++ when a is a prime number of 
the form 44 and — when it is of the form 44 + 1. 
+ 1, 
4. The expanded binomial (1 ~ 1)* 
n(n —1) (a — — 2) 
1.2.3 
~~ if thefe — be refpedtively multiplied by the feries 
25 334, Salle aa the - a except the nth, 
- whole cxprelfion li be ftill equal to zero. 
5. The continued bin 1.2. _ os 
=i—~n+ 
—1 
sen 8) + &c. = 03 
. # 1s exprefled 
by the formula n" — 2 (2 — 1)" + m(o— 2) (na — 2)" — 
oD “2 (n — 3)" + &e. 
6. cae _ number is either divifible by 5, or will 
ae ol a eoeneed plus or minus 1 
number is either divifible by 7, or ie leave 
alfo he Cue ode + 13; and generally we hav ° 
- - 19m, Orion +1 
IIn,ortlin+ 1 - - 7 
13m, Or 23a + 1 
et eee a 
= = 7H ojo tt 
AS 5m orsnta = - 
2 = = IIn, or rin +1 
x® Tm OTa4 i = 13m or ign tI 
a - - = 
a - - = 171, or iga +4 
x = 
I3#,orizga-+ 1 
which formule ar 
a) ie acho 
of ex raction 
are very convenient for determining whether 
umber be a complete power without the trouble 
eory of Numbers, where he will find a great v: 
{uch properties, an their — applications to aoe, 
sat ge yale trigonometry, &c. 
NoumBERS, for the Vous of charaGerifing, fee NoTA- 
TION. 
For that of expreffing or reading thofe already charaGerifed, 
fee NUMERATION. 
haha for the Meafure of a. See Measure 
Numser, Golden, in Chronolo M.C efines the 
me oak by the number of years elapfed fince that 
which had moon o ft day ; as that of the 
the new 
year 1500, whofe _ number was 0; which he takes 
for his ei ocha. 
the fame se s, though not precifely in the fame hour and 
Haase aa the day ; but within an hour and a half of the 
tn ares ch fenfe, golden number amounts ° a = with 
what we otherwife call /unar a or Meton 
Hence this period, called by the Ga neater, 
is not perfe@ly juft; there being a a or leap, at 
the end of each 312 years; i.e. in that time the lunations 
fali eut a day come. than the golden ‘cae exprefles 
This, a pe ie things, was what engaged pope Gre- 
gory XIII. t rm the calendar, to throw out the golden 
number, ad Tabititute the cycle of epaéts inftead of it, ee 
