RATIO. 



fraftions between - and 



And in the fame manner we may interpolate fourteen 

 ^^l; twenty-four between ^^ 



100 ' 100 



9208 9208 16149 



and - — ; and iix between and - . 



2931 2931 24339 



Having thus explained the nature of the operation, we 

 ftiall enter Icfs into detail in the following example. 



Example 2. — Accordiug to M de la Caille, the folar 

 year is 365'' 5'' 48™ 49', and confequently longer by 

 5'' 48™ 49' than tlie common year of 365 days. If this 

 difference were exaftly 6 hours, it would make one day at 

 the end of four common years : but if we wi(h to know 

 exaftly at the end of how many years this difference will 

 produce a certain number of days, we muft feek the ratio 



between 24'' and 5'' 48"' 49 , which we find to be — - — , 



20929 



fo that at the end of 86400 common years, we muff in- 

 tercalate 20929 days, in order to reduce them to tropical 

 years. 



Now as the ratio of 86400 to 20929 is expreffed in very 

 high terms, let it be required to find ratios in lower terms, 

 as near this as poffible. 



For this purpofe, we muft perform upon thefe numbers 

 the fame operations as in the preceding cafe ; thus : 



4 29 33 



&c. 



From which quotients we derive the following converging 

 fractions, ■u'tz, 



16 I 1 15 



2704 2865 5569 86400 



It appears farther, that as the fradions — , — , ,, , 



17^ 



are alternately lefs and greater than the fraftion , or 



' ° 20929 



— — , the intercalation of i day in 4 years would be 



5" 48'" 49' 



too much, of 7 days in 29 yearK too little, of 8 days m 

 33 years too much again, and fo on ; but each of thefe in- 

 tercalations will be the moft exaft, that it is poffible to make 

 in the fame fpace of time. 



Now if we arrange in two feparate feries, the fraftiont 

 that are lefs, and thofe that are greater, than the given 

 fraftions, we may infert or interpolate between certam of 

 thofe fraftions, as in the preceding examples. 



Taking firll thofe fraftions that are lefs than the given 

 one, and their correfponding (juotients, we (hall have, 



I I 15 



4 33 2865 86400 



I ' 8 ' 694 ' 20929* 



Hence it appears, that the only interpolation that can be 



c n- 2865 , 86400 



performed is between the two fractions ^ - and • ; 



*^ 694 20929 



which will admit of 14 intermediate fraftions ; which being 

 fupplied, as in the former example, gives the following 

 feries of converging fraftions, each lefs than the fraftion 

 originally propofed, viz. 



33 161 2865 8434 14003 



8' 39' 694' 2043' 3392' 

 19572 25141 30710 36279 

 6090' 7439' 

 52986 58555 



4 

 J 

 I 



474' 

 41848 

 10137' 

 69633 



16882' 



8788' 

 6A124 

 12835' 14184' 15533' 

 75262 80831 86400 

 19580' 



18231 19580' 20929 



And as the laft fraftion is the fame as the given fraftion, it 

 is evident that this feries cannot be carried farther : hence, 

 if we choofe to admit thefe intercalations only in which the 

 error is too much, the fimpleft and moft exaft will be thofe 

 of I day in 4 years, or of 8 days in 33 years, or of 39 in 

 161 years, and fo on. 



Let us now confider the decreafing fraftions : 



16 I 



7 

 29 



> 



7 



3 

 128 



2704 



5569^ 

 1349" 



31 (-ss 



Here it appears, that we may place 6 fraftions before the 

 firft, 2 between the firft and fecond, 15 between the fecosd 

 and third ; but between the third and fourth no fuch frac- 

 tion can be inferted. 



Thefe interpolations being made, vre ftiall have the fol- 

 lowiBg feries of decreafing fraftions, •viz. 



655' 694' 1349' 20929' 



Now we fee from the above fraftions, that the fimpleft 

 intercalation is that of one day in four common years, which 

 is the fonndation of the Julian calendar ; but that we Ihould 

 approximate with more exaftnefs, by intercalating only 

 7 days in the fpace of 29 common years, or 8 in the fpace 

 of 33 years, and fo en. 



148' 187' 226' 265' 304' 



I4I6 



