FIG 



FIG 



FIFE, in music, is a sort of wind in- 

 strument, being 1 a small pipe. 



FIFTEENTH, an ancient tribute, or 

 lax, laid upon cities, boroughs, See. through 

 all England, and so termed, because it 

 amounted to a fifteenth part of what each 

 city or town had been valued at ; or it 

 was a fifteenth of every man's personal 

 estate, according to a reasonable valua- 

 tion. In doomsday-book, there are cer- 

 tain rates mentioned for levying this tri- 

 bute yearly; but since, any such tax can- 

 not be levied but by parliament. 



FIFTH, in music, one of the harmoni- 

 cal intervals or concords. The fifth is 

 the second in order of the concords, the 

 ratio of the chord that affords it is 3 : 2. 

 It is called a fifth, as containing five 

 terms or sounds between its extremes, 

 and four degrees, so that in the natural 

 scale of music it comes in the fifth place 

 or order from the fundamental. The an- 

 cients called this fifth diapente. The im- 

 perfect and defective fifth, called by the 

 ancients 'semi-diapente, is less than the 

 fifth by a lesser semitone. 



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, 

 &c. 



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 

 continually the terms of any one, the suc- 

 cessive sums are the terms of the next 

 order, beginning from the first order, 

 which is that of equal units 1, 1, 1, 1, &c. ; 

 then the 2d order consists of the succes- 

 sive sums of those of the first order, form- 

 ing the arithmetical progression 1, 2, 3. 4, 

 &c. ; those of the 3d order the successive 

 sums of those of the 2d, and are the trian- 

 gular numbers 1, 3, 6, 10, 15, &.c. ; those 

 of the 4th order are the successive 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 1, 1, 1, 1, 1, &C. 



2. Arithmetical....!, 2, 3, 4, 5, &r. 



3. Triangulars 1, 3, 6, 10, 15, &c. 



4. Pyramidals.......!, 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 dif- 

 ferent sorts of triangular numbers, being 

 formed from an arithmetical progression, 

 whose common difference is 1. But if 

 that common difference is 2, the succes- 

 sive 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 num- 

 bers, or hexagons, and so on. Thus : 



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 1 those of the 

 preceding 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 

 nnits in the number which denominates 

 the order of figurates required, and divid- 

 ing those products always by the first 

 product : thus the triangular numbers 

 are found by dividing the products 1x2 ; 

 2x3; 3>X4, & c ' each by the first product 



1X2 : the first pyramids by dividing the 

 products 1 X 2 X 3 ; 2 X 3 X 4, &c. by 

 the first 1x2x3. And in general, the 

 figurate numbers of any order n are found 

 by substituting successively 1, 2, 3, 4, 5, 

 &.c. instead of z. in this general expression 



s X S +,t_Jil+2x.- + 3.&c. , 



1 X 2 X 3 X 4, &c. 



the factors in the numerator and denomi- 

 nator are supposed to be multiplied to* 

 gether, and to be continued till the num- 

 ber in each be less by 1 than that which 

 expresses the order of the figurutes re- 

 quired. See Simpson's Algebra. 



FIGURE, in physics, expresses the SUF- 



