and the Remainder f 8 Term of the Quotient a Sub-Multiple of the first Term of a 우: : then suppose is to be Divided by ,& =s. Again, lupposé s to be Divided by a Sub-Multiple thereof =b Prime come and giving the firt Term of the Quotient ja Sub-Multiple of the firt Term of , and then being Divided by ñ suppose the Remainder=o. I lay is = the greateft common Divisor required. to 2 .b 8 b wi and * i.e. The value of his g g b 2 Measures. { }axiom{1}}} ; Š and consequently «. Again, i "Measures box therefore Measures b.—: But aa . Measures and consequentlyr; therefore : Measures b tr=b. Whence'is certain that is a common Measure to, or of b and c. b Now in order to prove that it is the Greaceft, let w be the greatest common Measure of b and c. Then f being Prime sob, and w Measuring b; f will there fore с fore be * Prime to w; and fand d, 25. 7. Eucl, El. • Measuring c; wf will e, a consequence from the Measurec, os w will Measure a : 36 and 37. 7 Eucl. El. Bac Measures bor; there f fore we will Measuré bar, and consequently b - br=r. Again,g being ‘Prime to to and w Meafuring Fi -; g will therefore be «Prime' to w, and 'g and to Measuring ; gw wille Measure r, or 'w will Measure : But -: But Measures jas; -s; therefore w will Mea8 fure - - - ; and “confequentlyg - Again, 6 being 'Prime to, and w Measuring 8 8 b will therefore te “Prime tow; and hand w Measuring s;b w will 'Measures; viz. w will Measure bw ; w ; ; confe į quently w cannot exceed ; therefore they are equal. Finally, it is evident, that the like Demonstration with this may be applied to any other Number of Quantities. 2. E. D. Q. 8 ; Se&. 3. To abbreviate or bring a Frational Quan tity into its lowest Denomination. 1 Hule. Divide the Numerator and Denominator severally by their greatest common Divisor, and the respective Quotients plac'd Fraction-wise, is the Fraction required. Examples 44с Examples. 1. Let it be required to reduce to its leart Terms. od aac being Divided by c (which c is the greatest common Divi: for of aac and de), gives aa for a new Numerator; and dc being Divided by c, gives d for a new Denominator : consequently, na is the Fraction required. d bdc 2. In like manner at is found to be =* bc 4+d. In such fingle Fra&tions as ibee, the common Divisors (if there be any) are easily discover'd by inspection only: But in compound Fras dions it muft (for the most Pari) be found by our Lemma thus, 3. Let it be required to reduce aa + 2bat bb to its lowest bba Terms. First I Divide the greater of the two given Quantities, vix That which has the Greatest power of a in', (in respect of which Letter the Terms were plac'd) by the Less thus, nat aba + bb) a} - bba' (a - 26 a3 + abaat bba 2baa 2bbs - 2baa 4bba +2bba +267 Now since the firft Term of the Divisor has a greater Power of a in it than the first Term of 2bba + 263; this therefore is a Remainder: But the first Term thereof, viz. 2bba is not a SubMultiple of aa, the first Term of the said Divisor; wherefore I Divide the said Remainder by 2bb a Sub-Multiple thereof, Prime to the said Divisor, and giving the first Term of the Quotient a Sub-Multiple of the first Term of the aforesaid Divisor; thus 2bb) abba + 163 (a + b 2bba 263 2 263 o E And And the Quotient is a +b; by which I Divide the said Divisor ; thus atb) ant abat bb la tb Whence +b is the greatest common Measure required, by which the given Fraction will be reduc'd to its lowest Terms; chas +b) aa + 2ba + bb (a + b the new Numerator. aat ba bat bb bat bb stb) 03 - bba (as - be the new Denominator, a3 + ban baa bla bba ba A4 Consequently is the Fraction required. x4 4. In like manner will be reduc'd to its lower Terms thus, ** - 64) * _ j bb fx - for x64 - bb) -*} b6+ *b+ Remainder. ** -xxbb Fxbe bb) xxbb – b* (** - bb) * -- xbb (x xxbb xbb Remainder. Hence xx - bb is the greatest common Divisor of x* x4 be and xs - *b, by which will be reduc'd to xx+66 xs by this Sc&t. s. Ler it be required to reduce - ** + cx +-*- to its axti lowest Terms. First I Divide - **+tox-c by ** – 2x + 1, thus - + +c XX – 2x + 1) C (-! X--1 +1 *x + 2x – And the Remainder is + to But the first Term of this Remainder ; viz. FX is not aSubMultiple of the first Term of the foregoing Divifor ; to wit of xx; wherefore I Divide the said Remainder by its Sub-Mulmultiple + which is Prime to the said Divisor, and will give the firft Term of the Quotient a Sub Multiple of the first Term of the said Divisor ; thus, a |