f g a Sub-Multiple of the firft Term of r then fuppofe to be Divided by and the Remainder s. Again, fuppofés to be Divided by a Sub-Multiple thereof { = Prime to, and giving the firft Term of the Quotient g 44 a Sub-Multiple of the firft Term of and then b g g being Divided by —, suppose the Remaindero. I fay is the greateft common Divifor required. Demonftration. b C Measuress; therefore Measures 7 and confequently c. Again, Measures br +r=b. Whence'tis certain that is a common Measure to, or of b and c. Now in order to prove that it is the Greatest, let w be the greatest common Measure of b and c Then fbeing Prime to b, and w Measuring b; fwill there fore f C fore w will Measure br, and confequently b b→→→ -- ¿—r=r. Again, g being*Prime to, and Measuring ig will therefore be "Prime to w, and 'g and Measuring r; gw will Meafurer, or w will Meafure: But Measures -s; therefore w will Meag fure g ➡s; and 'confequently Again, b being 'Prime to, and » Measuring g g b will therefore be Prime to w; and b and w Measuring s ; b w will Measure s ; viz. w will Measure; confe quently w cannot exceed ; therefore they are equal. Finally, it is evident, that the like Demonftration with this may be applied to any other Number of Quantities. Q E. D. Sect. 3. To abbreviate or bring a Fractional Quantity into its lowest Denomination. Kule. Divide the Numerator and Denominator feverally by their greateft common Divifor, and the refpective Quotients plac'd Fraction-wife, is the Fraction required. Examples Examples. aac 1. Let it be required to reduce to its leaft Terms: cd aac being Divided by c (which c is the greatest common Divifor of aac and de), gives aa for a new Numerator; and de being Divided by c, gives d for a new Denominator: confequently, aa is the Fraction required. d A bdc 2. In like manner a + is found to be =+ bc a+d. In fuch fingle Fractions as thefe, the common Divifors (if there be any) are easily discover'd by inspection only: But in compound FraAtions it must (for the moft Part) be found by our Lemma thus, 3. Let it be required to reduce aa+2ba + bb Terms. a3 bba to its loweft First I Divide the greater of the two given Quantities, viz. That which has the Greateft power of a in't, (in respect of which Letter the Terms were plac'd) by the Lefs thus, aa+rba + bb) a3 — bba (a — 2b Now fince the firft Term of the Divifor has a greater Power of a in it than the firft Term of 2bba + 2b3; this therefore is a Remainder: But the firft Term thereof, viz. abba is not a SubMultiple of aa, the firft Term of the faid Divifor; wherefore I Divide the faid Remainder by 2bb a Sub-Multiple thereof, Prime to the faid Divifor, and giving the firft Term of the Quo tient a Sub-Multiple of the firft Term of the aforefaid Divifor; thus And the Quotient is a +b; by which I Divide the said Divifor; thus a+b) aa+2ba+bb (a+b aat ba ba + bb ba + bb And the Remainder is o Whence ab is the greateft common Measure required, by which the given Fraction will be reduc'd to its loweft Terms; thas a+b) aa +2ba + bb (a + b the new Numerator. a+b) a3bba (aa ba the new Denominator. x4 Hence xx bb is the greateft common Divifor of x4-4 and xx3 b', by which xx+bb x3 bb by this Sect. 5. Let it be required to reduce lowest Terms. First I Divide · xx + cx + x − c xx+cx+x-c by xx-2x+1, thus to its But the firft Term of this Remainder; viz. + - I X is not aSubMultiple of the firft Term of the foregoing Divifor; to wit of xx; wherefore I Divide the faid Remainder by its Sub-Mul +o multiple which is Prime to the faid Divifor, and will give the - I firft Term of the Quotient a Sub Multiple of the first Term of the faid Divifor; thus, |
