NUMBERS. 
284 284 284 284 284 . 
ey ee 7 7 
una 22° =, 222 a MO wy 2 yg, 27 a OO 
2 + 5 t+ to tO 22 + (20 
220 229 220 220 
+ Ge 7 55 + to + Gao = 7+ 
Figurate numbers are all thofe that fall under the general 
expreffion 
et4) Ae t2) (243) a 
3 &... 
and fier are ae to be of the hi 2d, 34, &c. order, ac- 
cording as m = 1, 2) 3, kc.: 
General term. 
Nat. feries, 1, 2; 35 45 5» &c. = n 
I 
xft ord. 1, 3, 6, 10, 15, &c. se+%) 
n(n+1) (n+ 2) 
ad ord. 1, 4) 10, 20, 35, &c. 
n(n 
tu 2s @ 
gd ord. ty 5) 15; 352 70 &c. ame Gs?) ez 8) 
ath ord. 1, 6, 21, 56, & 
Thefe are  otherwif called pyramidal numbers 
Polygonal numbers are the fums of differ ent and inde- 
pice pene cal feries, and are termed natural or lineal, 
triangular, ante ed or als Fs hae hexagonal, &5c. 
feri are 
numbers, according from which they 
generated. 
ineal, or natural — are formed from the fucceflive 
{ums ofa ferics of units ; 
Unit 1, 1,1, 1, &e. 
Nat. Cubes : : 3> 4s 52 6, &c. 
General form, a, 
Triangular numbers are oe fucceflive fums of an hase 
feries, ie ae = with unity, the common difference of 
which i hus 
re feri S85 - 92335 4, 5> &e. 
caer numbers, . 3, 6, 10, 15, &c. 
2 
— th 
a 
General form, 
Quadrangular or fquare numbers are the fucceflive fums of 
an arithmetical feries, ae with unity, the common 
difference of which is 
Arithmetical feries ¥y 39 5) 7s 9, &e. 
Quadrangular or fare numbers, I, 4, 9, 16, 25, &c. 
General form, 
= 2’. 
2n*—on 
2 
Pentagonal numbers are the fucceffive fums of an arith- 
metical feries, beginning with unity, the common difference 
of which is 3 ; 
perenne feries, 1, 4, 7, 10, 13, &e. 
Pentagonal numbers, 1, ie 12, 22, 35, &c. 
32 
General form, 
and fo on for hexagonal, agonal ae ae 3 the general 
form for the m-gonal feries of numbers being 
(m — 2) n*— pas 
= . 
Thefe are called high aie el a they fiiay 
be always arranged in rm eral geometrical 
eae after os they are pleat See Pory- 
AL Number 
BERS are oe divided into ab/elute, abftract, con- 
crete, di Palas seth rae omogencal, rational, irrational, 
ae A ead for i ee . e refpettve ioe es. 
tion of the feveral properties, forma, tivifor oduéts, 
f integral numbers, is fubjeQ wa indeed confidered 
by fome o mathematici 
f{perfed with many ma rginal notes of his sai and which 
may be confidered as containing the firft ger our prefent 
theory. Thefe were afterwards conse aa by 
the celebrated Fermat, in his edition of the fame work, pub- 
lifhed after his death i 70, W a oar the 
ct 
o 
np 
prepariiig a treatife on the theory of aes which would 
contain * multa varia et abftrufiffima nuamerorum lpr 
but unfortunately this work never appeared, and moft of his 
oo remained without demonftration ne a confiderable 
tim 
°. times. The former, befides what is contained i 
his “ Elements of Algebra,” and his ‘ Analyfis Infinito- 
,’ has feveral pape erfourgh Ads, i 
which are given the demonitrations of many of Fermat’s 
heorems. has ring on this fubje 
8 contained in chap ‘“‘ Meditationes Algebraicz.’’ 
Grange, din th of 
tions to Euler’s ‘Kigebra, It is, however, but pine that 
this branch of analyfis has been reduced into a regular fy{- 
tem; a tafk that was firft performed by Le Gendre, in his 
“< Effai fur la Théorie des Nombres,”” Paris, 1800; a fecond 
and nea t the 
nes m efe 
eee difplay | the talents - their refpeCtive au 
contain a complete developement of this interefting th cory. 
T e latter, in particular, has opened a new field of inquiry, 
by the application of the a eda of numbers ise ne folu- 
tion of binomial equations of the form, *”—1= n the 
folution of an h depends the ‘ivifion of the circle into a 
(See Potyeon.) rlow, of the Royal Military 
Academy, has alfo publifhed a concife treatife on this fub- 
je@t, entitled « An elementary Inveftigation of the Theory 
£ Numbers ;”’ to which work we are indebted for many of 
the preceding remarks and definitions, as alfo for feveral of 
the following properties of numbers, in which we have 
generally omitted the demonftrations, as thefe would have 
carried us beyond the limits prefcribed to the prefent 
article. 
Properties 
