CALCULUS. 



dimimition any one of tViefe magnitudes mud undergo, 

 in oi-der to have to the other ;iny nmltiphcate or fuhmul- 

 tipHcate ratio of thefe magnitudes in tlieir given Itate ; 

 or any fuch ratio of tliem, as is denoted by fraftions or 

 fiirds ; or (co fpeak ftili more generally) a ratio, which has 

 to the ratio of the firll-nTrntioned of tliefe magnitudes, to 

 the other the ratio of any two magnitudes whatever, of the 

 fame but of any kind. Neither have I been able to find, that 

 any other author lia-i (liewn geometrically, in a general way, 

 ji when any number of r^/ioj- are to be comp'.ninded or de- 

 compounded with a given ratio, how much either of the 

 magnitudes in tlie given rati'j is to be augmented or dimi- 

 nifhed, in order to have to the other a ratio, wliich is equal 

 to the given ratio, compounded or decompounded with the 

 other ratios. To invclbgate all thefe geometrically, and to 

 i fix general laws in relation to them, is the objeft of this 

 paper ; which, as it treats of a fubjcft as ujw as it is general, 

 . I flatter myfelf will not prove unacce[)table to this learned 

 focicty. It would be altogether fuperfluous for me to men- 

 tion the great advantages that mull neced'arilv accrue to ma- 

 thematics in general, from an accurate invelligation of this 

 fubjeft, fince its influence extends more or lefs to every 

 branch of abflraft fcience, when any data can be alcertain- 

 ed for reafoning from. I (liall in a fubfequent paper take 

 an opportunity of fiicvving, how from the theorems after- 

 1 wards delivered in this, a method of realoning with fniite 

 magnitudes geometrically may be derived without any con- 

 fjderation of motion or velocity, applicable to every pur- 

 pofe, to which fl ixions have been applied." He then de- 

 fines magnitude to be " that wir.cli admits of increafe or 

 decreafe," and quantity to be " the degree of magnitude," 

 obferving, that " by magnitude he means, befide cxtenfion, 

 every thing which admits of more or lefs, or what can be 

 increafedordiminilhed, fuch as n7//oj-, velocities, powers, &c. 

 In a fliort performance of his, printed in Latin, in 1776, 

 entitled " Leges Metaphyfiert, feu Principia Mathematica, 

 iqus: Omnia fere ad Ma2;nitndmum Rationcs, Rationumque 

 iRelationes fpedlautia univerfahter gubernant et indefinite 

 proferunt," he givcs the fame metaphyfical definitions of 

 magnitude and quantity, with a declaration of what he 

 means by magnitude, viz. 



" Magnitudo eft id, quod augeri vel diminui potefl." 

 " Qiiantitas ell gradus magnitudlnis." 

 , " Per magnituduRm, prxterextenfionem, omnia, qua; au- 

 geri vel diminui poiTunt, ficut ratioues, velocitates, vires, &c. 

 intelligo." 



And in his Unlverfal Comparifon itfelf he fays, " When 

 r fpeak of magnitude, I mean to be underftood as taking it 

 in its general, abftrafted, unlimited, and metaphyfical accep- 

 tation, viz. to be whi\tcver adujits of mure or lefs, of increafe 

 or decreafe;. and by quantity I mean the degree of magni- 

 tude." And to prevent any mifconception of his meaning 

 or intention, in regard to his mode of expreffing magnitudes 

 geometrically, and connefting tliem together by means of 

 the cudomary figns, he makes ufe of the following woids : 

 — " When I mention the ium of magniti;des, or fpeak of 

 them as additive or fubtraftive, I only mean that they are 

 to be taken with or from other magnitudes of the fame kind. 

 The figns plus and minus are only intended to denote fuch 

 aggregates and difTcrcnces, or to coiuieft magnitudes to- 

 gether in thefe relations, and by no means to convey any 

 numerical ideas in the following theorems or formula;, or to 

 imply in themfclves any fort of myllery, or even inean- 

 ing, independent of the magnitudes thus connedkd. By 



. C-D ^ A-B . A-BV , , , , 



A. — 7::—' A. — TT— ' A.- — :?; — ' Sit. I mean a fourth prO' 



portioual to D, C-D, and A; a fourth proportional to 

 B, A-B, and A ; a fourlli proportional to B, A — B, and 

 , A-B 

 ^' — jT — ' &c. rtfpci'tively. I am obliged to have rc- 



courfe to fuch eX])refrions in the formulx, which ought not 

 on any account to be confidered as algebraic, as it is impof- 

 lible to cxprefj them by means of geometrical fehemcs or 

 hguies in fuch a way, as to be fufficienlly or clearly under- 

 ilood." 



As both the differential and flnxionary calculi are immc- 

 duitely, and with the greateil facility derivable from this 

 calculus wlien the exprelfions in it are fuppofed to become 

 numerical, or the llandard of comparifon to be I, or arith- 

 metical unity, and in forms too alcogether unexceptionable, 

 and as it is itfelf chiefly derived from the ill and 5th formu- 

 Ix in theorem 5 of the univerfal comparifon, it is perhaps 

 nectfl^ny to prove the trutli of thefe formuke, by means of 

 wliat is delivered in the paper firil herein above-mentioned, 

 that was read before the Royal Society of Loud n in 1777, 

 in which, among other things, it is demonllrated geometri- 

 cally, that if n be any whole or integral number whatfo- 

 ever, and A and B be any two homogeneous magnitudes^ 



A-B . n-i «_2 . A-B'-' 



^ + - 



A.- 



A, 



+ &c. to 



A 



A-B 



B ' I 1 '" ]i 



has to B, fuch a multiplicate ratio of A to 



B, as is expreffed by the number n. Now if in this geome- 

 trical expreffion for A, we fubflitute its equal B-}-A— B 



// — — — - " n — 1 

 we get the following exprelTion : B -\ A — B-|- -• • 



B 



+ ■ 



»— I 



2 



n-2 A-B 



B^ 



-1- &c. to 



A^n? 



hav- 



B'-" 



ing alfo to B fuch a multiplicate ratio of A to B as is ex- 

 preffed by the number h. This expreffion may alfo be eafily 

 derived geometrically by means of Kuchd's elements, in the 

 manner purfued by i\'Ir, Glcnie in thnt paper. For let MN, 

 NO, and OP be ea^h equal to, or reprefent B, and let NR 

 reprefent A. Let OP, NR be drawn perpendicularly to 

 VO, or otherwife, if in the fame angle j and let the redlangles 

 or parallelograms MR, NP be compleatcd. Let LNI be 

 a fourth proportional to OP, MN, and NR-OP, and lef 

 the rctlangle or paralleiograin LQ^be compleated. 



Then (14. E. 6.) LT is equal to TR, and fince MT h 

 equal to Tcj^by conlhuftion, LM is equal to QR, or A-B, 



a-d LN to A or B -f A-B. And lince by {13. E. 6.) 

 MR has to NP the ratio compounded of the ratios 



Z 



R 



A I 



H 



A-B 



lA—t! + 



aTITI' 



B. A-B 



B 



or 



A-B 



A-B 



A-B 

 Q. 



D 



VOL.V, 



B 



B 



V K 



M F 



5A 



N B 



O 

 of 



