VAN 



liis writings were fo abfurd or blafphemous as to entitle lum 

 to the chai'after of a defpifer of God and religion." An 

 Apology for V^anini was publifhed in Holland in 17 12, by 

 Peter Frederick Arp, a learned lawyer. Morcri. Mojlieim, 

 vol. V. 



VANISHING Fractions, are fraftions in wliich, by 

 giving a certain value to the variable quantity or quantities 

 entering into them, both numerator and denominator become 

 zero, and confequently the fraftion itfelf is then i. 



The idea of fraftions of this kind firft originated about 

 the year 1 702, in a conteil between Varignon and RoUe, 

 two French mathematicians of fome eminence, concerning 

 the principles of the differential calculus, of which the latter 

 was a ftrenuous oppofer ; and amongft other arguments 

 againft the truth of the doftrine which had then been re- 

 cently introduced, he propofed an example of drawing a 

 tangent to a certain curve, at the point where the two 

 branches interfeft each other ; and as the fradlional expref- 

 fion for the fubtangent, according to that metliod, had both 

 its numerator and denominator equal to zero, or o, he re- 

 garded fuch a refult as abfurd, and adduced it as a proof 

 of the fallacy of this mode of folution. But the myilery 

 ■was foon after explained by Jolm Bernouilli ; and upon a 

 renewal of the difpute, ftill farther by Saurin, who (hewed 

 that the fraftion in the cafe here mentioned had a real value. 

 Thefe fraftions were alfo the caufe of a violent controverfy 

 between Waring and Powell in 1760, when thefe gentle- 

 men were candidates for the mathematical profefTorniip 

 at Cambridge : Waring maintaining that the fraction 



, when a; is i , is equal to 4 ; and Powell, or rather 



Maferes, who is commonly fuppofed to have condufted the 

 difpute on the part of the latter, that it was equal to o, or 

 indeed that it could have no value whatever : and it mull 

 be acknowledged that the fame diiference of opinion relative 

 to this kind of fraftions ftill exifts in all its force. Wood- 

 houfe, in his " Principles of Analytical Calculation," in 

 treating of thefe quantities, after affuming the fimple cafe of 



— to find the value of it, when x = a, obferves, that 



the fignificatioa of this expreffion is, that .x" — aS is to be 

 divided hy x — a, and the refult of that divifion is .v -|- a, or 

 putting .V = a, it becomes a + a, or 2 a. This refuh, how- 

 ever, he remarks, is no diredl and natural confequence arifing 

 from the principles of calculation, but, on the contrary, it 

 is a refult arbitrarily obtained, by extending a rule, and ob- 

 ferving a certain order in the procefs of calculation. 



To the queftion, what does 



become when it = a; 



the obvious and logical anfwer is , and the queftion 



° a — a 



is, whether in this form it will admit of any further reduc- 

 tion. It is true, if we operate upon this quantity accord- 

 ing to the rules laid down in other apparently fimilar cafes, 



we obtain "—^ — ■ = a+ a = 2a; but here is evidently an 



a — a 

 extenfion given to a rule beyond what was firft intended ; 

 for this rule was inftituted for operating on real quantities, 

 whereas in this cafe we have employed it on quantities 

 having no value whatever, being in fad the divifion of o by 

 O, for which abftraftedly no rule can be given. This, how- 

 ever, is not a cafe peculiar to thefe fraftions. It is to the 

 Vol. XXXVI. 



V A N^ 



fame fourcc we muft attribute the introduftion of the 

 negative fymbol, and all the myfteries attendant upon it, as 

 well as to every kind of imaginary quantity. 



In vol. i. p. 219, of Bonnycaftle's Treatife of Algebra, 

 we have the following rule for finding the value of vanilhing 

 fraftions. 



1. If both the terms of the given fraftion be rational, 

 divide each of them by their greateit common mcafure ; then, 

 if the hypothefis which is found to reduce the original ex- 

 preffion to the form J be apphed to the refult, it will give 

 the true value of the fraftion under confidcration. 



2. Wlien any part of the fraftion is irrational, obferve 

 what the unknown quantity is equal to, when the numerator 

 and denominator both vanifli, and put it equal to that 

 quantity + and — i ; then if this be fubftituted for the un- 

 known quantity, and the roots of the furds be extrafted to 

 a fufficient number of places, the refult, when i is put equal 

 to o, will give tlie true value of the fraftion. From which 

 rule the author obtains the following refulls ; viz. 



I. ■ ■ — = za, when x — a. 



^=. \b, when .v 



4. ::::: m a" ', when X = a. 



X — a 



See Bonnycaftle's Algebra, Woodhoufe's Principles of Ana- 

 lytical Calculation, and Barlow's Diftionary. 



VANISI, in Geography, a town of Turkiftl Armenia ; 

 2 1 miles W. of Akalzike. 



VAN-LAER, in Biography. See Bamboccio. 



VANLOO, Carlo, was the fon of an artift little known, 

 and was born at Nice in 1705. For fome time he refided 

 at Rome, and ftudied under Benedetto Luti. In 1723 he 

 went to Paris, where he gained the firft prize for an hif- 

 torical painting, and was employed with his elder brother, 

 John Baptifte Vanloo, in repairing the paintings of Pri- 

 maticcio, at Fontainbleau. In 1727 he again yifited Italy, 

 and afterwards pafied fome time at the court of Turin, 

 where he painted a feries of piftures from Taffo. On his 

 return to his native country in 1734, he was admitted into 

 the academy, the king conferred upon him the order of 

 St. Michael, and appointed him his principal painter ; and he 

 repaid thefe comphments by his affiduity and his abihty. 

 He had acquired by his ftudies in Italy more correftnefs 

 than his countrymen generally poffefled, and he certainly 

 prevented the French fchool from running farther into the 

 aff'efted ftyle of Cgypel and De Troyes, and yet liis ftyle 

 can only be called loofe and mechanical, with little relifti of 

 the higher beauties of art. He died in 1765, at the age 

 offixty. 



Vanloo, Mad., the daughter of Somis, the great vie 

 linift of Turin, concert-mafter to the king of Sardinia, 

 and wife to the celebrated painter Vanloo of Paris, was 

 born in 1710, and in 1726 ftie was thought the beft 

 finger of her time. We have feen a beautiful print of 

 Mad. Somis, from a painting of Vanloo previous to her 

 marriage. After her nuptials, ftie fettled at Paris, and was 

 living there in 1 754. The firft wife of the elder Cramer, 

 the excellent performer on the violin, and leader of a band, 

 was a daughter by tliis marriage ; which accounts for her 

 good tafte and captivating manner of finging to her own 

 accompaniment on the pedal harp. She was a moft ac- 

 4 G complifhed 



