C A L 



Computilb were by tlic lawyers caUed cj'i:iloi:ts, when I'licy 

 were cither ihncs, or newly frctil men ; thole of a better 

 condition were denominated rtiktilalons, or tiiinu-rnni : or- 

 dinarily tliere was one of thefe mailers in each family of 

 didindion ; the title of wliofc office was a calciihs, or a 

 r.Uioiubus. 



The Roman judges anciently irave their opinions by 

 ca'culi, which were while for abfolutioo, and black for con- 

 demnation. Hence calculus alhiis, in ancient writers, de- 

 notes a favom-able vote, cither in behalf of a perfon to be 

 ahfolved and acquitted of a charge, or eleifted to fome dig- 

 nity or poll ; as calculus iiiger did the contrary. This ufage 

 •is fuid to have been borrowed from the Thracians, who 

 marked their happy or profperous days by white, and their 

 \iiihappy by black pebbles, put each night into an urn. 

 Hence alfo the pliiafe8,_/^'/;rtrf, iiotare aliquid albo, ti:^rove 

 lap'illo feu culcuhi. 



Befidesthc diverfity of colour, there were fome alfo which 

 had figures or charadlers painted or engraven on them ; 

 Bs thole whicii were in ufe in taking the fiiffiages both in 

 the feiiate and at allcmb'ies of the people. The letters 

 marked upon thefe calculi were U. R. for iit'i rogas, and A. 

 for anliquo ; the firll of wliich exprelfcd an approbation of 

 the law, the latter a rtjcftion of it. Afterwards the judges, 

 who fat in capital caufcs ufed calculi marked with the letter 

 iK.ior ahf/jlvo,C for coiukmuo, and N. L. fur uou liquet ; 

 iignifying a more full information was required. 



" We may alfo mention another fpecies of calculi ufed at 

 the public games, whereby the rank and order in which the 

 athlctx were to fight were determined. If for inllance they 

 were twenty, then twenty of thtfe pieces were caft into an 

 urn ; each ten were marked with numbers from one to ten, 

 and the law was, that each of thofe who drew, (liould fight 

 him who had drawn the fame number. Thefe were called 

 calculi alUetici. 



Calculus is alfo ufed in ^ucieiil Grammatic IVriters for a 

 kind of weight equal to two grains of cicer. Some make 

 it equivalent to the ftliqua, which is tqual to three grains of 

 barley. Two calculi made the ceratium. 



Calculus alfo denotes a certain method of performing 

 mathematical inveftigations and refolntions. Thus, we fay 

 the antecedental calculus, the arithmetical or numeral cal- 

 culus, the algebraical calculus, the calculus of derivations, 

 the differential calculus, the exponential calculus, the fluxi- 

 onal calculus, the integral calculus, the literal or fymboli- 

 Cal calculus, 5cc. See the following articles. 



Calculus, Antcceclental. This is a geometrical method 

 of reafoniug without any confideration of motion or velo- 

 city, appUcable to every purpofe, to which fluxions have 

 been or can be apphed. It was invented by James Glenie, 

 efq. A. M. and fellow of the Royal Societies of Loudon 

 .and Edinburgh, as early as 1774, but not publifhed till the 

 year 179.3, when it was printed in London for G. G. Land I. 

 Robinfons, Paternoller-row. As he derived it from an exami- 

 nation of the antecedents of ratios, having given confequents 

 and 3 given llandard of comparifon in the various degrees 

 of augmentation and diminution, which they undergo by 

 compofition and decompofition, he denominated it the An- 

 tecedental Calculus. It is certainly founded on principles 

 admitted into the very firll elements of geometry, and re- 

 peatedly made ufe of by Euclid himfelf. The inventor of 

 this calculus does not in imitation of modern mathematicians 

 talk of the powers of magnitudes and quantities, but con- 

 fiders every expreflion in it as truly and llriAly geometrical, 

 or rather as univerfally metrical in a geometrical form. For 

 although the general formula in his univerfal comparifon, 

 from which it is derived in an eafy, fimple, direft, and con- 



C A I. 



cife mnnner, are ellabliflied on the f'.me prir.e'ptes witli 

 thofe of Euclid's elements of geometry, and are deduced in 

 a demonftrative way from an application of fome of the 

 truths contained m thofe elcmenls to the doilrine of ratios 

 or proportion, they are, ftrictly fpeaking, generally metrical, 

 and extend not only to geometry, but alfo to algebra and 

 arithmetic, when the llandard of comparifon i.s fuppofed to 

 become i or arithmetical unity. The tranfitions from them 

 in tiieir geometrical forms to their algebraic and numerical 

 ones, are as Mr. Glenie has obfervcd, fo natural, fo fcientific, 

 and fo beautiful, that they cannot fall to hirnidi the mind 

 with the higheft pleafure and fatisfadtion, in pointing out, 

 as it were, at one general view the connexion between thefe 

 diflerent fcienccs, and unfolding the reafons of llieir various 

 operations, from the fame indifputable and mathematical 

 principles. He has not, like other malhem:itieiaiis, in con- 

 iidering ratio confined himfclf to any particular modification 

 of it ; but lie has regarded it as a magnitude poffciriug all 

 the meafurable alFcclions ol any other magnitude, viz, ad- 

 dition, fubtrafliou, multiplication, divifion, and ratio or 

 proportion, or, as admitting, like other inagniiudes, of aug- 

 m.entation or diminution, of incie:'.fe or decreafe. He is the 

 firll perfon, that we know of, who has roundly and expreff- 

 ly conlidered ratio or proportion as an afie<ili">n of ratio, 

 whic'.i It nii'll unqueftionahly is. And by applying the ele- 

 ments of geonietiy to this confideration of it, he has ex- 

 tended the geometrical analyiis indefinitely farther than any 

 one before him, eitlier among the ancients or moderns, as 

 far as we are able to difcov.-r, has carried it. That he 

 formed the defign of applying geometry to ratios under this 

 conception of them, as magnitudesyj/;^ftt(;-;j, cayable of all 

 the poflible degrees of increafe and dccreale, though they 

 know no divtrfity of dimenfion in refpeft of kuid. being ho- 

 mogeneous, and differing not in kind, but only in degree, 

 and atlually made fuch an application of it when he could 

 not have been above eighteen or twenty years of age, is evi- 

 dent from his own performances. For in a fhort introduc- 

 tion to a paper read before the Royal Society of London, 

 the 6th of March, 1777, and publillied in tlieir tranfaclions, 

 which is entitled, " The general Mathematical Laws, that 

 regulate and extend Proportion univerfally ; or a Method of 

 comparing Magnitudes of any kind together, in all the 

 poffible Degrees of Increafe and Decreafe," he makes ufe of 

 the follovving words : 



" The doflrine of propoition laid down by Euclid and 

 the application of it given by him in his elements, form the 

 bafis of almoll all the geometrical reafoniug made ufe of by 

 mathematicians both ancient and modern. But the reafon- 

 iug of geometers with regard to proportional magnitudes 

 have fcldom been carried beyond the triplicate ralio, which 

 is the proportion that fimilar fohds have to one another, 

 when relerred to their homologous linear dimenfions. This 

 boundary however comprehends but a very limited propor- 

 tion of univerfal companion, and almoil vanidies into no- 

 thing when referred to that endlefs variety of relations, 

 which mud neceflarilv take place between geometrical mag- 

 nitudes in the infinite poffible degrees of increafe and de- 

 creafe. The firft of thete takes in but a very contrafted 

 field of geometrical comparifon ; whereas the lall extends it 

 indefinitely. Within the narrow compafs of the firll, the 

 ancient geometers performed wonders ; and their labours 

 have been pulhed ftill farther by the ingenuity and indefa- 

 tigable induftry of the moderns. But no author, that L 

 have been able to meet with, gives the leaft hint or in- 

 formation with regard to any general method of ex. 

 prefling geometrically, when any two magnitudes of the 

 fame kind are given, what degree of augmentation or 



6 diminution 



