VOL. XLTtll.] PHILOSOPHICAL TRANSACTION'S. 305 



//. An Account of a Book, intitled, P. D. Pauli Frisii* Mediolanensis, &c. 



Disquisitio Mathematica in Causam Physicam Figurte et Magnitudinis Telluris 



Nostrce. Milan 1752. By Mr. J. Short, F. R. S. p. 5. 



It may be laid down as a rule in mixed mathematics, " That the determination 

 of no physical quantity be carried further than the observations, or other mecha- 

 nical measures, can bear;" lest there follow this incongruity, of the conclusion 

 being more extensive than the premises. It would be absurd, for instance, in 

 the resolution of a triangle, to compute an angle to the exactness of seconds, 

 or a side to centesms of an inch, when perhaps the instruments used can mea- 

 sure no angle less than 10 minutes, or a side only to the exactness of a foot. 

 The conclusions of arithmetic and geometr}' are indeed rigorously true, but they 

 are only hypothetical; and whenever the quantities, that enter any practical 

 question, can only be measured within certain limits, it would be in vain to look 

 for an answer perfectly accurate. The error of the instrument becomes itself one 

 of the data, and we must content ourselves to find the limits which the quantity 

 sought cannot well exceed, or fall short of, by such rules as Mr. Cotes has 

 given in his excellent treatise on the subject. 



In like manner, when any physical theory is deduced from observations, its 

 accuracy will still be in proportion to that of the observations on which it is 

 founded. Sir Isaac Newton, in computing the ratio of the earth's axis to its 

 equatorial diameter, confines himself to a reasonable approximation, and to 3 

 places of figures (229 to 230) ; because, whether that ratio is deduced from the 

 difterent lengths of isochronous pendulums in different latitudes, or from the 

 measurement of distant degrees of a meridian, or from both, the elements of 

 the calculus can scarcely furnish a greater degree of exactness. And of the same 

 judicious caution, we have many other examples in the works of that incompa- 

 rable author. On the other hand, when observations and theories are brought 

 together and compared, nothing can be justly inferred against a theory from its 

 disagreement with the observations, unless that disagreement is greater than can 

 be fairly imputed to the imperfection of instruments, and to the unavoidable 

 mistakes of an observer; especially if the difference should be sometimes in 

 excess, and at other times in defect ; or, as some of the observations should en- 

 tirely vanish. 



Though these rules, manifestly well-founded, have been followed by all the 

 best writers, our author observes, that several ingenious men, both in France 

 and in Italy, have deviated from them, particularly in treating of the famous 

 question concerning the figure o'f the earth. Some, with Messrs. Clairaut and 

 Bouguer, attributing too much to the observations that have been made, and 



* Fail] Frisi was born at Milan about the year 1789. ' 

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