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SECTION XIII. 
OF INDETERMINATE, OR UNLIMITED PROBLEMS. 
A Problem is said to be indeterminate, or unlimited, 
when the equations, expressing the conditions 
thereof, are fewer in number than the unknown quan 
tities to be determined; such kinds of Problems, strictly 
speaking, being capable of innumerable answers: but 
the answers in whole numbers, to which the question is 
commonly restrained, are, for the general part, limited 
to a determinate number; tor the more ready discover 
ing of which, 1 shall premise the following 
LEM MA. 
Supposing to be an algebraic fraction, in its 
lowest terms, x being indeterminate, and a, b, and c, 
given whole numbers; then, I say, that the least inte 
ger, for the value of x that will also give the value of 
aX - an integer, will be found by the following 
method of calculation. 
Divide the denominator fcj by the co-efficient (a) of 
the indeterminate quantity: also divide the divisor by the 
remainder, and the last divisor, again, by the last re 
mainder ; and so on. till an unit only remains. 
Write down all the quotients in a line, as they follow; 
under the first of which icrite an unit,and under the second 
write the first; then multiply these two together, and 
having added the first term of the lower line (or an unit) 
to the product, place the sum under the third term of the 
upper line : multiply, in like manner, the next two cor 
responding terms of the two lines together, and, having 
added the second term of the lower to the product, put 
down the result under the fourth term of the upper one.: 
proceed on, in this way, till you have multiplied by every 
number in the upper line.