I N D 
to each other; an inftance of which are thofe two curious 
ones, propofed by M. Fermat, as a challenge to all the 
mathematicians of Europe, viz. iff. To find a cube num¬ 
ber which added to all its aliquot parts {hall make a fquare 
number; and 2d, To find a fquare number which added 
to all its aliquot parts {hall make a cubic number; which 
problems were anfwered after feveral ways by Dr. Wallis, 
as well as fome others of a different nature. See the 
. Letters that paffed between Dr. Wallis, the lord Broun- 
ker, fir Kenelm Digby, &c. in the doctor’s works ; and 
the works of Fermat, which were collefted and publifhed 
by his fon. Moft authors on algebra have alfo treated 
more or lefs on this part of it, but more elpecially Kerfey, 
Preftet, Ozanam, Kirkby, &c. But afterwards, mathema¬ 
ticians feemed to have forgotten fuch queftions, if they 
did not even defpife them as ufelefs, when Euler drew 
their attention by fome excellent compofitions, demon- 
ftrating fome general theorems, which had only been known 
by induction. M. la Grange has alfo taken up the fub- 
je£t, having refolved very difficult problems in a general 
way, and difcovered more direft methods than heretofore. 
The fecond volume of the French tranflation of Euler’s Al¬ 
gebra contains an elementary treatife on this branch, and, 
with la Grange’s additions, an excellent theory of it ; 
treating very generally of indeterminate problems of the 
firft and fecond degree, of folutions in whole numbers, of 
the method of indeterminate coefficients, &c. 
Finally, Mr. John Leflie has given, in the fecond volume 
of the Edinburgh Philof. Tranfaftions, an ingenious pa¬ 
per on the refolution of indeterminate problems, refolving 
them by a new and general principle. “ The doctrine of 
indeterminate equations,” fays Mr. Leflie, has been fel- 
dom treated in a form equally fyftematic with the other 
parts of algebra. The folutions commonly given are 
devoid of uniformity, and often require a variety of af- 
fumptions. The object of this paper is to refolve the 
complicated expreffions, which we obtain in the folution 
of indeterminate problems, into Ample equations, and to 
do fo, without framing a number of affumptions, by help 
of a Angle principle, which, though extremely Ample, ad¬ 
mits of a very extenlive application. Let A X B be any 
compound quantity equal to another, C X D, and let m 
be any rational number affumed at pleafure; it is manifeft 
that, taking equimultiples, A X m B =2 C X m D. If, 
therefore, we fuppofe that A =2 m D, it mull follow that 
C 
vi B = C, or B 2=—. Thus two equations of a lower di- 
m 
menfion are obtained. If thefe be capable of farther de- 
compofition, we may affume the multiples n and p, and 
form four equations ftill more Ample. By the repeated 
application of this principle, an higher equation, admit¬ 
ting of divifors, will be refolved into thofe of the firft or¬ 
der, the number of which will be one greater than that 
of the multiples affumed.” 
For example, refuming the problem at Aril given, viz. 
to And two rational numbers, the difference of the 
fquares of which lhall be a given number. Let the 
given number be the produfl of a and b ; then by hy- 
pothefls, x 2 — y 2 — ab ; but thefe compound quantities 
admit of an eafy refolution,Tor x + y X x —y = a x b. 
If therefore we fuppofe x + y 22 m a, we fhall obtain 
x — y 222 —; where m is arbitrary, and, if rational, x 
m 
and y muff alfo be rational. Hence the refolution of thefe 
two equations gives the values-of xandy, the numbers 
r • ,. • vi 2 a 4- b 
{ought, m terms of m ; viz. x = --, and y 222 
m 2 a—b ' 
zm 
INDETERMINATELY, adv. Indefinitely; not in 
any fettled manner.—His perfpicacity difeerned the load- 
ftone to refpect the north, when ours beheld it indetermi¬ 
nately. Brown. 
INDETERMINATION, /. Want of determination ; 
I N D 7 
want of Axed or ftated direction. — By contingents I under¬ 
hand all things which may be done, and. may not be done, 
may happen, or may not happen, by reafon of the 2 ^de¬ 
termination or accidental concurrence of the caufes. Bram- 
hall. * 
INDETER'MINED, adj. Unfettled ; unfixed,..—We 
fhould not amufe ourfelves with floating words of indeter- 
mined Agniflcation, which we can ufe in ieveral fenfes to 
ferve a turn. Locke. 
INDEVIL'LARS, a town of France, in the department 
of the Doubs: three quarters of a league eaft of St. Hypo- 
lite, and two fouth-eaft of Blamont. 
INDEVOTION, f. Want of devotion; irreligion.— 
Let us make the church the feene of our penitence, as of 
our faults; deprecate our former indevotion, and, by an ex¬ 
emplary reverence, redrefs the fcandal of profanenefs. 
Decay of Piety. 
INDEVOUT', adj. Not devout; not religious ; irreli¬ 
gious.—He prays much ; yet curfes more; whilfl he is 
meek, but indevout. Decay of Piety. 
IN'DEX,/ [Latin.] The dilcoveter ; the pointer-out. 
—That which was once the index to point out all vir¬ 
tues, does now mark out that part of the world where the 
lealt of them refldes. Decay of Piety. —The hand that 
points to any thing, as to the hour or way.—They Jjave 
no more inward felf-confcioufnefs of what they do or 
Buffer, than the index of a watch of the hour it points to. 
Bentley. —The table of contents to a book.—If a book has 
no index or good table of contents, ’tis very ufeful to 
make one as you are reading it, and in your index to take 
notice only of parts new to you. Watts. 
In fuch indexes, although fmall 
To their fubfequent volumes, there is feen 
The baby flgure of the giant mafs 
Of things to come, at large. Shakefpeare , 
Taulman compared a book without an index to a. 
warehoufe without a key, or an apothecary’s drawer 
without a label. 
Index, in anatomy, denotes the fore-Anger. It is thus 
called from . indico, I point or direct; becaufe that Anger 
is generally fo ufed : whence alfo the extenfor indicis is 
ca.\\e&indicator. 
Index of-a Globe, is a little flyle Atted on to the north 
pole, and turning round with it, pointing to certain divi- 
iions in the hour-circle. It is fometimes alfo called 
gnomon. 
Index in arithmetic and algebra, otherwife called the 
exponent, is the number that fliows to what power it is 
underftood to be raifed: as in io 3 , or a 3 , the flgure 3 is 
the index or exponent of the power, flgnifying that the 
root or quantity 10 or a is raifed to the 3d power. See 
Exponent, vol. vii. 
I ndex is the fame with what is otherwife called the 
charaEleriftic, ox exponent, of a logarithm ; being that which 
fliows of how many places the abfolute or natural number 
belonging to the logarithm conflfts, and of what nature it 
is, whether an integer or a fraction; the index being lefs 
by 1 than the number of integer-figures in the natural 
number, and is pofitive for integer or whole numbers, 
but negative in fractions, or in the denominator of a 
fraction; and, in decimals, the negative index is 1 more 
than the number of ciphers in the decimal, after the point, 
and before the firft fignificant figure; or, ftill more gene¬ 
rally, the index fliows how far the firft figure of the na<- 
tural number is diftant from the place of units, either 
towards the left hand, as in whole numbers, or towards 
the right} as in decimals; thefe oppolite cafes bein°- 
marked by the correfpondent figns -J- and —, of oppofne 
affections, the fign — being fet over the index, and not 
before it, becaufe it is this index only which is under¬ 
ftood as negative, and not the decimal part of the Wa- 
ritlun. Thus, in this logarithm 2-4234097, the figilres 
of whofe natural number are 2651, the 2 is the index, 
and, being pofitive, it fhovvs. that the firft figure of the 
number. 
