FIG 
FIG 
FIG, the fruit of the ficus, or fig-tree. See 
Ficus. 
FIGURAL, or Figurate numbers, are 
such as do or may represent some geome- 
trical figure in relation to which they are 
always considered, as triangular numbers, 
pentagonal numbers, pyramidal numbers, 
At. 
FIGURATE numbers, are distinguished 
into orders according to their place in the 
scale of their generation, being all pro- 
duced one from another, viz. by adding con- 
tinually the terms of any one, the successive 
sums are the terms of the next order, begin- 
ning from the first order, which is that of 
equal units 1, 1, 1, 1, &c. ; then the 2d 
order consists of the successive sums of those 
of the first order, forming the arithmetical 
progression 1, 2, 3, 4, &c. ; those of the 3d 
order the successive sums of those of the 2d, 
and are the triangular numbers 1, 3, 6, 10, 
15, &c. ; those of the 4th order are the suc- 
cessive sums of those of the 3d, and are the 
pyramidal numbers 1, 4, 10, 20, 35, &c. ; 
and so on, as below. 
Order. Name. „ Numbers. 
1. Equals i, 1, l, l, 1 , &c. 
2. Arithmetical... 1, 2, 3, 4, 5, &c. 
3. Triangulars 1, 3, 6, 10, 15, &c. 
4. Pyramidals 1, 4, 10, 20, 35, &c. 
5. 2 d Pyramidals.. 1, 5, 15, 35, 70, &c. 
6. 3 d Pyramidals. .1, 6, 21, 56, 126, &c. 
7. 4 th Pyramidals. 1, 7, 28, 84, 210, &c. 
The above are all considered as different 
sorts of triangular numbers, being formed 
from an arithmetical progression, whose 
common difference is 1. But if that com- 
mon difference is 2, the successive sums will 
be the series of square numbers ; if it be 3, 
the series will be pentagonal numbers, or 
pentagons ; if it be 4, the series will be 
hexagonal numbers, or hexagons, and so on. 
Thus : 
Arithmetical. 
1st:. Sums or Polygons. 
Cd. Sums, or 2d. Polygons. 
1, 2, 3, 4, 
1, 3, 5, 7, 
1, 4, 7, 10, 
lj 5, 9, Joj 
Tri. 1, 3, 6, 10, 
Sqrs. 1, 4, 9, 16, 
Pent. 1, 5, 12, 22, 
Hex. 1, 6, 15, 28, 
1, 4, 10, 20, 
1-, 5, 14, 30, 
1, 6, 18, 40, 
1, 7, 22, 50, &c. 
And the reason of the names triangles, 
squares, pentagons, hexagons, &c. is, that 
those numbers may be placed in the form 
of these regular figures or polygons. The 
figurate numbers of any order, may be 
found without computing those of the pre- 
ceding order, which is done by taking the 
successive products of as many of the terms 
of the arithmeticals 1, 2, 3, 4, 5, &c. in 
their natural order, as there are units in the 
number which denominates the order of 
figurates required, and dividing those pro- 
ducts always by the first product : thus the 
triangular numbers are found by dividing the 
products 1 x 2 ; 2 x 8 ; 3 x 4,&c.each by 
the first product 1x2: the first pyramids 
by dividing the products 1X2X3;2X3X4, 
Sec. by the first 1X2X3. And in general, 
the figurate numbers of any order n are found 
by substituting successively 1, 2, 3, 4, 5, 
Ac. instead of s in this general expression 
z_X » + 1 X a + 2 X z + 3, &e. 
1 
where 
X 2 X 3 x 4, &c. 
the factors in the numerator and denomina- 
tor are supposed to be multiplied together, 
and to be continued till the number in each 
be less by 1 than that which expresses the 
order of'tiie figurates required. See Simp- 
son’s Algebra. 
FIGURE, in physics, expresses the sur- 
face or terminating extremities of any body ; 
and considered as a property of body af- 
fecting our senses, is defined a quality which 
may be perceived by two of the outward 
senses. Thus a table is known to be square 
by the sight, and by the touch. 
Figures, in arithmetic, are certain cha- 
racters whereby we denote any number 
which may be expressed by any combina- 
tion of the nine digits, &c. See Digit. 
Figure, in botany, a property of natural 
bodies, from which marks and distinctive 
characters are frequently drawn. Figure is 
more constant than number ; mbre variable 
than proportion and situation. The figure 
of the flower in the same species is more 
constant than that of the fruit: hence Lin- 
mens advises to arrange under the' same 
genus such plants as agree invariably in the 
flowers, that is, in the calyx, petals, and 
stamina, although the fruit or seed-vessel 
should be very different. The seed-vessels 
of the different species of French honey- 
suckle, wild senna, acacia, Syrian mallow, 
and sopliora, are exceedingly diversified in 
point of figure. Hence some former bota- 
nists, who paid more attention to the parts 
of the fruit, considered many of these spe- 
