RATIO. 



1416 1577 1738 1899 2060 



"343^' 382' 42> ' 460' 499' 



2221 2382 2543 2704 5569 

 7P"' 577' 616' 65s' 1349' 



which are all lefs than the propofed fradion, and exprcfTed 

 ui Icfs terms ; and each of which is nearer than any other 

 tradion tliat can be exprclfed in Icfs terms. 



Hence we conclude, that if we only attend to the inter- 

 calations in which the error is too fmall, the fimplell and 

 mod fxad are thofe of i day in 5 years, of 2 days in 



9 years, of 3 days in 13 years, of 4 days in 17 years, and 



10 on. 



In the Gregorian calendar, only 97 days are intercalated 

 in 400 years ; but it is evident, from the preceding table, 

 that it would be much more exaft to intercalate 109 days 

 in 450 years. 



But it mud beobferved, that, in the Gregorian reforma- 

 tion, the determination of the year given by Copernicus was 

 made ufe of, which is 365'' 5'' 49' 20" ; and fubllituting 



this inftead of the fraftion 



86400 

 20929' 



we (hall have 



86400 

 29060' 



feries, 



rather -- - ; whence we may find, by the preceding me- 



131 

 thod, the quotients 4, 8, 5, 3 ; and from them the prin- 

 cipal fraftions 



4853 



4 33 il9 545 



i' 8' 41' 131' 



which, except the two firft, are quite different from thofe 

 before determined. However, we do not find amongft 



thefe fraftions - -, which is that adopted in the Gregorian 



97 

 calendar ; and this fraftion cannot even be found among the 

 interpolated fraftionc, which might be inferted in the two 



— , — , and ^, ^ : for it is evident that it 

 I 41 8 131 



could only be between the laft two fraftions, between 

 which, becaufe of the number 3, (the correfponding quo- 

 tient,) there can be but two fraftions interpolated, which 



are , and — -- : whence it appears, that it would have 



49 90 



been more exaft, if, in the Gregorian reformation, they 

 had only intercalated 90 days in the fpace of 371 years. 



If we reduce the fraAion , fo as to have for its 



37 



numerator the number 86400, it will become — ^- , which 



20952 



ellimates the tropical year at 365'' 5'' 49' 12". 



In this cafe, the Gregorian intercalation would be quite 

 exact ; but as obfervations fhew that the year is fliorter 

 than this by more than 20", it is evident that, at the end of 

 a certain period of time, we muft introduce a new inter- 

 calation. 



If we adopt the determination of de la Caille, it follows, as 



the denominator 97 of the above fraftion, viz. — , lies be- 



97 

 tween the denominators of the fifth and fixth principal 



fraftions already found, that, from what has been dated 



above, the fraflion will be nearer the truth than 



39 



. But as aflronomers are ftill divided with rctrard to 

 97 ^ 



the exaft length of the year, we (liall refrain from giving a 

 decifive opinion on this fubjeft. For more on the rcdudtion 

 of ratios, fee Lagrange's Additions to Euler's Elements 

 of Algebra. 



Ratios, Prime anil Ultimate, is a fpccics of computation, 

 which we owe to the fertile genius of Newton. The an- 

 cients, in order to extend the geometry of riglit lines to 

 curvilinear figures, had recourfe to the method of cxhauf- 

 tions, in which they made ufe of what is called the reduSiu 

 ad abfurdum method, which, though logical, is extremely tedi- 

 ous, and to avoid which, Cavalcrius propofed his method of 

 indivifibles, piibliflied in 1635 under the title of " Geome- 

 tria Indivifibilibus," in which he was followed by Dr, 

 Wallis and others of the 17th and i8th centuries. In this 

 method every line was fuppofed to coniilt of a number of 

 other lines indefinitely fmall ; every curve was confidered as 

 a polygon of an indefinite number of fides, each fide be- 

 ing indefinitely fmall ; a fohd was fuppofed to confift of 

 an infinite number of plane feftions, or of indefinitely thin 

 laminre, and fo on ; fuppofitions which in many inilances 

 led thofe, who adopted them, into errors and inconfiftencies, 

 which indeed it was very difficult to avoid. 



To obviate both the tedioufnefs of the ancients, and the in- 

 accuracy of the moderns, Newton introduced his method of 

 prime and ultimate ratios, the foundation of which is con- 

 tained in the firft lemma of the firft book of his " Prin- 

 cipia." Many difficulties have been ftarted, and much con- 

 troverfy concerning the proof of it ; all of which would 

 have been avoided, had either the author or his readers ob- 

 ferved, that he is in reality laying down the definition of a 

 term, viz. being ultimately equal, and not proving a propo- 

 fition. Taking, therefore, this firft lemma for a definition, 

 it may be illuftrated as follows. 



Let there be two quantities, one fixed and the other vary- 

 ing, fo related to each other, that, I ft, the varying quantity 

 continually approaches to the fixed quantity ; and, 2dly, that 

 the varying quantity never reaches or paftes beyond that which 

 is fixed : 3dly, that the varying quantity approaches nearer 

 to the fixed quantity than by any affigned difference. Then 

 is fuch a fixed quantity called the limit of the varying quan- 

 tity ; or, in other words, the varying quantity may be faid 

 to be ultimately equal to the fixed quantity, Thefe three 

 conditions may be expreffed more diftintlly thus. i. The 

 difference between the varying quantity and the fixed quan- 

 tity muft continually decreafe. 2. This difference muit 

 never become either nothing or negative. 3. This differ- 

 ence muft become lefs, in refpeft of the fixed quantity, 

 than by any afligned ratio ; or the difference between the 

 two quantities muft become a lefs part of the fixed quantity 

 than any fratlional part that is affigned, however fmall the 

 fraftion exprefling fuch part may be. Wherever thefe pro- 

 perties are found, the fixed quantity is called the limit of the 

 varying quantity, or the varying quantity is laid to be ulti- 

 mately equal to the fixed quantity. The laft cxpreffioii, 

 however, muft not be underftood in its ftrift hteral fenfe, 

 there being no ultimate Jlate, no particular magnitude, that 

 is the ultimate magnitude of fuch a varying quantity. Under 

 the word quantity in this definition, muft be included not 

 only numbers, lines, &c., but more efpecially ratios con- 

 fidered as a peculiar fpecies of quantity ; but as the con- 

 10 fideration 



